Transfronteriza pre-service teachers managing,resisting, and coping with the demands of mathematical discourse

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.     

WISE Choices? Understanding Occupational Decision-making in a Climate of Equal Opportunities for Women in Science and Technology

Since the launch of the Women into Science and Engineering (WISE) campaign in 1984, many initiatives to increase the participation of women in these areas of work have been launched under its banner and the WISE approach has come to represent the dominant discourse on equal opportunities for women in science and technology, having a major influence on both policy and practice. This article examines the WISE discourse in depth, arguing that WISE has had only very limited success because it is so narrowly focused on women's 'choices', which it understands as being constrained both by a lack of information about scientific and technological work and by a masculine image of science and technology which, it infers, is alienating to women. Drawing on empirical research which examined both women's and men's occupational decision-making processes, this article takes issue with this construction of the problem, arguing that whilst the assumptions of the WISE discourse cannot be supported empirically, the discourse itself nevertheless continues to structure and limit the space women have to speak of the conflicts and contradictions they experience, explanations for which require a better understanding of the ways in which subjective experiences of both gender and sexuality impinge upon work choices.     

Mathematical images in advertising: constructing difference and shaping identity, in global consumer culture

Mathematics educators have long emphasised the importance of attitudes, beliefs, and feelings towards mathematics, as crucial in motivating (or not) its learning and use, and as influenced in turn by its social images. This paper is about images of mathematics. Our search for advertisements containing such images in UK daily newspapers, during 2006–2008, found that 4.7 % of editions included a “mathematical” advert, compared with 1.7 % in pilot work for 1994–2003. The incidence varied across type of newspaper, being correlated with class and gender profiles of the readership. Three quarters of advertisements were classified as containing only very simple mathematics. ‘Semiotic discursive’ analysis of selected advertisements suggests that they draw on mathematics not to inform, but to connote qualities like precision, certainty, and authority. We discuss the discourse on mathematics in advertising as ‘quasi-pedagogic’ discourse, and argue that its oversimplified forms, being empty of mathematical content, become powerful means for regulating and ‘pedagogising’ today's global consumers.     

Spoken Language and Mathematics

Teachers and learners use spoken language in the three part mathematics lesson advocated by the National Numeracy Strategy (NNS). This document recognises the importance of language by emphasising 'the correct use of mathematical vocabulary' in the raising of standards. Research into the use of vocabulary in science suggests that the use of scientific words does not necessarily demonstrate conceptual understanding. Pupils and teachers appear to ascribe different meanings to scientific words because of their relative positions on the novice/expert continuum. To explore whether mathematical words could cause similar problems, data collected from six groups of teachers and learners was subjected to discourse analysis to provide evidence of how spoken language was used and how meaning and understanding were achieved. The implications of the findings on the use of language by both teachers and learners in the NNS mathematics lesson, including vital areas for reflection by teachers, are fully considered.     

小学数学双语教学中的课堂教学语言选择

小学数学双语教学是以外语为语言背景进行的教与学的交流,其重点在于解决数学问题的同时解决语言问题。因此,教师在处理教学交流时应注意不同交流方式中教学语言的选择,其中,常规性交流适宜用纯英语进行,相关性交流尽量要使用外语,程序性交流应该最大程度地使用外语,构思性交流适宜从使用母语向使用外语逐步过渡,同时,小学双语数学教学要注重学生对语言与数学结合的思维习惯的培养。     

Hating school, loving mathematics: On the ideological function of critique and reform in mathematics education

Students’ engagement with fictions in the form of “word problems” plays an important role in classroom practice as well as in theories of mathematical learning. Drawing on the Dutch historian Johan Huizinga and the Austrian philosopher Robert Pfaller, I show that this activity can be seen as a form of play or game, where it is pretended that mathematics is useful in real life in a way that it is not. With Pfaller, I argue that play can take hold of the imagination of the players, infusing everyday life with meaning borrowed from the imagery of the play and that these effects are more powerful when the play is forced and takes an institutionalized form. I show that mathematics education does in fact have these characteristics, including sophisticated mechanisms for translating in-game performance (test scores) to real-life goods (grades and examinations). A central theme of the article is the perceived discrepancy between mathematics education as it is, and how it supposedly could and should be in light of the properties of mathematics. The analysis implies that this gap actually is an effect of play and thus an inherent property of mathematics education itself.     

Engineering Difference: Discourses on gender,sexuality and work in a college of technology

ABSTRACT This article explores the reasons behind the continuing underrepresentation of women in science and engineering. Reviewing existing debates, the article argues that there exists an impasse in debates on women in engineering which can be related to the dualistic frameworks within which existing research has tended to work. Explanations for the poor representation of women in these areas tend to emphasise either individual or structural factors, neither of which allow women's agency to be fully understood. Drawing on empirical research which examined the subject choices and occupational decision-making processes of two groups of students in a college of technology (one undertaking a course in traditional 'women's work' and the other a course in a non-traditional area), the analysis here employs a post-structuralist approach, employing aspects of discourse theory, to offer a new understanding of women in engineering and of the relationship between gender, sexuality and work, more generally.     

Collaborative learning with multi-touch technology: Developing adaptive expertise

Developing fluency and flexibility in mathematics is a key goal of upper primary schooling, however, while fluency can be developed with practice, designing activities that support the development of flexibility is more difficult. Drawing on concepts of adaptive expertise, we developed a task for a multi-touch classroom, NumberNet, that aimed to support both fluency and flexibility. Results from a quasi-experimental study of 86 students (44 using NumberNet, 42 using a paper-based comparison activity) indicated that all students increased in fluency after completing these activities, while students who used NumberNet also increased in flexibility. Video analysis of the NumberNet groups indicate that the opportunity to collaborate, and learn from other groups' expressions, may have supported this increase in flexibility. The final phase of the task suggests future possibilities for engaging students in mathematical discourse to further support the development of mathematical adaptive expertise.     

Shifting Engagements in Figured Worlds: Middle School Mathematics Students' Participation in an Architectural Design Project

Project-based curricula have the potential to engage students' interests. But how do students become interested in the goals of a project? This article documents how a group of 8th-grade students participated in an architectural design project called the Antarctica Project. The project is based on the imaginary premise that students need to design a research station in Antarctica. This premise is meant to provide a meaningful context for learning mathematics. Using ethnography and discourse analysis, the article investigates students' engagement with the imaginary premise and curricular tasks during the 7-week project. A case study consisting of scenes from main phases of the project shows how the students took on concerns and responsibilities associated with the figured world proposed by the Antarctica Project and how this shaped their approaches to mathematical tasks (Holland, Lachicotte, Skinner, & Cain, 1998). Participating in the figured world of Antarctica and evaluating situations within this world was important for how students used mathematics meaningfully to solve problems. Curricular tasks and classroom activities that facilitated students in assuming and shifting between roles relevant to multiple figured worlds (i.e., of the classroom, Antarctica, and mathematics) helped them engage in the diverse intentions of curricular activities.     

Educational forms of initiation in mathematical culture

A review of literature shows that during the history of mathematics education at school the answer of what counts as ‘real mathematics’ varies. An argument will be given here that defines as ‘real mathematics’ any activity of participating in a mathematical practice. The acknowledgement of the discursive nature of school practices requires an in-depth analysis of the notion of classroom discourse. For a further analysis of this problem Bakhtin’s notion of speech genre is used. The genre particularly functions as a means for the interlocutors for evaluating utterances as a legitimate part of an ongoing mathematical discourse. The notion of speech genre brings a cultural historical dimension in the discourse that is supposed to be acted out by the teacher who demonstrates the tools, rules, and norms that are passed on by a mathematical community. This has several consequences for the role of the teacher. His or her mathematical attitude acts out tendencies emerging from the history of the mathematical community (like systemacy, non-contradiction etc.) that subsequently can be imitated and appropriated by pupils in a discourse. Mathematical attitude is the link between the cultural historical dimension of mathematical practices and individual mathematical thinking.     

Working on arithmetic word problems when English is an additional language

There has been little research in the UK into how students learning English as an additional language (EAL) learn mathematics. This article reports results from a three‐year study of the participation of learners of EAL in Year 5 in an arithmetic word problem task. The research, drawing on ideas from discursive psychology, used discourse analysis to explore patterns of attention in students' interaction as they worked in pairs or threes. The article briefly describes four patterns of attention: to genre, to mathematical structure, to narrative experience and to written form. Further analysis explored how students used attention as part of the social activity involved in working on the task. The rest of the article illustrates how students used attention to narrative experience to make links between word problems and their own experience, as well as to negotiate their relationships with each other.     

An Ode to Imre Lakatos: Quasi-Thought Experiments to Bridge the Ideal and Actual Mathematics Classrooms

This paper explores the wide range of mathematics content and processes that arise in the secondary classroom via the use of unusual counting problems. A universal pedagogical goal of mathematics teachers is to convey a sense of unity among seemingly diverse topics within mathematics. Such a goal can be accomplished if we could conduct classroom discourse that conveys the Lakatosian (thought-experimental) view of mathematics as that of continual conjecture-proof-refutation which involves rich mathematizing experiences. I present a pathway towards this pedagogical goal by presenting student insights into an unusual counting problem and by using these outcomes to construct ideal mathematical possibilities (content and process) for discourse. In particular, I re-construct the quasi-empirical approaches of six!4-year old students’ attempts to solve this unusual counting problem and present the possibilities for mathematizing during classroom discourse in the imaginative spirit of Imre Lakatos. The pedagogical implications for the teaching and learning of mathematics in the secondary classroom and in mathematics teacher education are discussed.     

Older Women Training to Teach

The relationship of age to gender in training, education and the wider society has been largely neglected. This article starts by exploring the discourses of ageism and sexism which impact on older women. The article then considers the situation for one group of older women who are all preparing to become teachers. Drawing on the perspectives of 11 older women, this article illuminates some of the dilemmas which mature trainee teachers experience. The intention in this article is to ensure that age is included when the complexity of women's oppression and discrimination is analysed.     

Integrating Case Studies in Engineering Mathematics: A response to SARTOR 3

Case studies have long been used to support the mathematics education of undergraduate engineers. Changes in the mathematical ability of entrants to engineering programmes and, indeed, the changing nature of many of the programmes themselves indicate the need to make the students' mathematical experiences more 'user friendly'. We describe here an approach which uses case studies, not as illustrations of applications of mathematics after a mathematical topic has been discussed, but in a fully integrated central role as vehicles for whole group discussion from which the students 'discover' the necessary mathematics which is taught subsequently. Not only is the 'carrot' of the application then central to their learning, but the need for the mathematics being taught is also clearly demonstrated. This approach has been tried with a group of 50 first year engineers. The effects on student motivation, ability and knowledge retention are discussed together with an indication of the Integrated Case Studies which were used.     

John Wilson and the Place of Morality in Education

This paper asks whether it would be better not to talk about morality in schools. The issue is raised through a consideration of changes in public discourse and especially in educational discourse, where categories such as ''personal, social and health education'' and ''citizenship education'' are more salient than ''moral education''. Drawing on John Wilson's arguments, the paper considers claims for the indispensability of the concept of morality. It is argued that such claims, in Wilson's own writings, are applied to both an ''individual'' and a ''social'' conception of morality. Contrary to Wilson, the paper argues that the ''wisest strategy'' for public education is to take the social conception of ''morality in the narrow sense'' as a central focus.     

To know and to teach: Mathematical pedagogy from a historical context

Historical texts tell us much about how mathematical concepts and techniques were conceived and evolved. But they can also tell us even more. Usually, their contents embody a pedagogy, specific organizational forms and methods used for teaching mathematics. This article isolates and examines several particular pedagogical techniques: the use of an instructional discourse; a logical sequencing of mathematical problems and exercises and employment of visual aids, evident in historical works. It concludes that much of present day mathematical pedagogy evolved from distant historical antecedents.     

When the Rules of Discourse Change,but Nobody Tells You: Making Sense of Mathematics Learning From a Commognitive Standpoint

The interpretive framework for the study of learning introduced in this article and called commognitive is grounded in the assumption that thinking is a form of communication and that learning mathematics is tantamount to modifying and extending one's discourse. These basic tenets lead to the conclusion that substantial discursive change, rather than being necessitated by an extradiscursive reality, is spurred by commognitive conflict, that is, by the situation that arises whenever different interlocutors are acting according to differing discursive rules. The framework is applied in 2 studies, one of them featuring a class learning about negative numbers and the other focusing on 2 first graders learning about triangles and quadrilaterals. In both cases, the analysis of data is guided by questions about (a) features of the new mathematical discourse that set it apart from the mathematical discourse in which the students were conversant when the learning began; (b) students' and teachers' efforts toward the necessary discursive transformation; and (c) effects of the learning–teaching process, that is, the extent of discursive change actually resulting from these efforts. One of the claims corroborated by the findings is that school learning requires an active lead of an experienced interlocutor and needs to be fueled by a learning-teaching agreement between the interlocutor and the learners.     

Learning by Seeing by Doing: Arithmetic Word Problems

"Learning by doing" in pursuit of real-world goals has received much attention from education researchers but has been unevenly supported by mathematics education software at the elementary level, particularly as it involves arithmetic word problems. In this article, we give examples of doing-oriented tools that might promote children's ability to "see" significant abstract structures in mathematical situations. The reflection necessary for such seeing is motivated by activities and contexts that emphasize affective and social aspects. Natural language, as a representation already familiar to children, is key in these activities, both as a means of mathematical expression and as a link between situations and various abstract representations. These tools support children's ownership of a mathematical problem and its expression; remote sharing of problems and data; software interpretation of children's own word problems; play with dynamically linked representations with attention to children's prior connections; and systematic problem variation based on empirically determined level of difficulty.