Undergraduate Students’ Conceptions of Mathematics: An International Study

In this paper, we report on an international study of undergraduate mathematics students’ conceptions of mathematics. Almost 1,200 students in five countries completed a short survey including three open-ended questions asking about their views of mathematics and its role in their future studies and planned professions. Responses were analysed starting from a previously-developed phenomenographic framework (Reid et al., 2003) which required only minor modification. Students’ conceptions of mathematics ranged from the narrowest view as a focus on calculations with numbers, through a notion of mathematics as a focus on models or abstract structures, to the broadest view of mathematics as an approach to life and a way of thinking. Broader conceptions of mathematics were more likely to be found in later-year students (p<0.001) and there were significant differences between universities (p<0.001). The information obtained from the study not only confirms previous research, but also provides a basis for future development of a monitoring questionnaire.     

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.     

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.     

我国小学数学教科书中数学史料的分析与建议

作为一门课程,越来越多的人开始从文化这一视角来关注数学,《全日制义务教育数学课程标准（实验稿）》也明确指出“数学是一种文化”。通过对我国小学数学教科书中数学史料的分析,我们发现,从数学文化的角度理解数学,从数学文化史的角度理解数学史料,在现行小学数学教科书中不仅体现得很不够,而且还存在简单化倾向等问题,即：对数学史料的理解单一、对数学史料内容的选择单一、对数学史料的编排单一等。对此,我们认为,应从文化史的角度来理解数学史料,从古今中外数学文化史中依据多种主题,选择“史料”内容,以“螺旋上升的”形式来编排所选择的主题“史料”内容。     

Building on student mathematical thinking in whole-class discourse: exploring teachers’ in-the-moment decision-making,interpretation, and underlying conceptions

Often, mathematics teachers do not incorporate whole-class discourse of students’ various ideas and solution methods into their teaching practice. Particularly complex is the in-the-moment decision-making that is necessary to build on students’ thinking and develop their collective construction of mathematics. This study explores the decision-making patterns of five experienced Dutch mathematics teachers during their novice attempts at orchestrating whole-class discourse concerning students’ various solution methods. Our goal has been to unpack the complexity of their in-the-moment decision-making during whole-class discourse through lesson observations and stimulated recall interviews. We investigated teacher decision-making adopting a model that combines two perspectives, namely (1) we explored student-teacher interaction with regard to building on student thinking and (2) we explored how the teachers based decisions during such interaction upon their own personal conceptions and interpretation of student thinking. During these novice attempts at orchestrating whole-class discourse, the teachers created many situations for students to articulate their thinking. We found that at certain instances, teachers’ in-the-moment decision-making resulted in opportunities to build on student thinking that were not completely seized. During such instances, the teachers’ decision-making was shaped by the teachers’ own conceptions of the relevant mathematics and by teacher conceptions that centered around student understanding and mathematical goals. Our findings suggest that teachers might be supported in their novice attempts at whole-class discourse by explicit discussion of the mathematics and of their conceptions with regard to student understanding and mathematical goals.

     

数学课程改革的核心环节 --更新数学教师的数学观

我国数学教育改革的现状、国内外数学教育的历史和数学教师数学观的独特性,都说明了研究数学教师数学观的重要性。改变数学教师的数学观是数学课程改革的必要条件。因此,数学课程改革的核心环节是更新数学教师的数学观。     

数学文化融于微积分教学的实践与思考

数学文化是大学生文化素养的重要组成部分,如何在大学数学教学中融入数学文化尤为重要。微积分作为变量数学中的重要内容,讨论其中蕴含哪些数学文化以及怎样在其教学中渗透数学文化,不仅能帮助学生很好的领悟微积分的基本概念和思想方法,而且有助于培养学生科学、理性的思维方式和交流表达能力,提高整体文化素养。     

Conceptions for relating the evolution of mathematical concepts to mathematics learning—epistemology, history, and semiotics interacting To the memory of Carl Menger (1902–1985)

There is an over-arching consensus that the use of the history of mathematics should decidedly improve the quality of mathematics teaching. Mathematicians and mathematics educators show here a rare unanimity. One deplores, however, and in a likewise general manner, the scarcity of positive examples of such a use. This paper analyses whether there are shortcomings in the—implicit or explicit—conceptual bases, which might cause the expectations not to be fulfilled. A largely common denominator of various approaches is some connection with the term “genetic.” The author discusses such conceptions from the point of view of a historian of mathematics who is keen to contribute to progress in mathematics education. For this aim, he explores methodological aspects of research into the history of mathematics, based on—as one of the reviewers appreciated—his “life long research.”     

The emerging of teachers' conceptions of new subjects inserted in mathematics programs: the case of informatics

The changes in mathematical curricula induced by the introduction of informatics in school represent the general framework of this research. In particular we focus on the teacher's role by analysing the different choices taken by mathematics teachers when faced with a curriculum reform induced by the introduction of informatics in secondary school courses (age 14–16). Our hypothesis is that these choices are the consequence of conceptions teachers have about informatics and its teaching in relation to the teaching of mathematics. Thus, through a case study research method, we focus on mathematics teachers' conceptions of informatics and its teaching. An attempt is made at outlining a typology of these conceptions, based on the different orientations identified.     

Parents' representations of their children's mathematics learning in multiethnic primary schools

There is a growing concern that governmental calls for parental involvement in children's school mathematics learning have not been underpinned by research. In this article the authors aim to offer a contribution to this debate. Links between children's home and school mathematical practices have been researched in sociocultural studies, but the origins of differences within the same cultural group are not well understood. The authors have explored the notion that parents' representations of school mathematics and associated practices at home may play a part in the development of these differences. This article reports an analysis of interviews with parents of 24 children of Pakistani and White origin enrolled in primary schools in England, including high and low achievers in school mathematics. The extent to which the parents represented their own school mathematics and their child's school mathematics as the ‘same’ or ‘different’ are examined. In addition, ways in which these representations influenced how they tried to support their children's learning of school mathematics are examined. The article concludes with reflections on the implications of the study for education policy.     

Consequences of Individual Differences in Children's Formal Understanding of Mathematical Equivalence

Experts claim that individual differences in children's formal understanding of mathematical equivalence have consequences for mathematics achievement; however, evidence is lacking. A prospective, longitudinal study was conducted with a diverse sample of 112 children from a midsized city in the Midwestern United States (M_age [second grade] = 8:1). As hypothesized, understanding of mathematical equivalence in second grade predicted mathematics achievement in third grade, even after controlling for second-grade mathematics achievement, IQ, gender, and socioeconomic status. Most children exhibited poor understanding of mathematical equivalence, but results provide clues about which children are on the path to constructing an understanding and which may need extra support to overcome their misconceptions. Findings suggest that mathematical equivalence may deserve more attention from educators.     

我们应该如何认识数学的本质--对林夏水先生"论数学的本质"一文的商榷

对数学的本质应该怎样认识，这不仅是数学哲学的一个基本问题，而且是一个具有时代性、前瞻性、发展性、综合性的数学哲学核心问题．对数学本质的认识不应该从传统数学哲学的角度退缩到方法论的一个狭隘的层面，而应该从更广阔的、更为多样的角度进行透视．林夏水先生的“数学是一门演算的科学”的见解，忽略了数学的非演绎性和非算法性，因而无法完全概括数学的本质．在数学深刻的思想与知识变革过程中，“数学是关于模式的科学”的观点是对数学本质的恰当定位。     

Affective productions of mathematical experience

In underscoring the affective elements of mathematics experience, we work with contemporary readings of the work of Spinoza on the politics of affect, to understand what is included in the cognitive repertoire of the Subject. We draw on those resources to tell a pedagogical tale about the relation between cognition and affect in settings of mathematical learning. Our interest is first captured in the way in which one teacher’s priority of establishing an inclusive learning community occasionally harboured what appeared to be pedagogically restrictive conceptions of mathematics. Yet, the classroom practices that produced these conceptions promoted the students’ motivation and provided meaningful access to mathematical learning within the classroom collectivity. In a second example, the postponement of scientific encapsulation in bodily imitations of planetary movement kept alive a shared dynamic sense of an elliptical orbit. In both of these cases, we draw on Spinoza’s work to show how the affectivity of classroom practice constituted conceptions of cognition and of mathematical activity crucially linked to the imperatives of participation.     

数学教育心理学的发展与"数学素质"的再认识

当前对于数学素质的认识有多种理解,从数学教育心理学历史发展来看,这些认识有其历史渊源,即可以看作是基于行为连接观、认知建构(个体)观和社会建构观。新理论对原有理论从来都是一种继承关系,而不是一种替代关系。实践中的“怀旧”和研究上的“追新”实际上割裂了理论的历史联系。就理论的历史逻辑来看,无论用什么时期的视角来理解和实践“数学素质”都有其合理性,关键在于不同的理论适用于不同的教学目标和内容。     

Would the Real Lakatos Please Stand Up

Although Imre Lakatos described the work published in his book Proofs and Refutations as a study of mathematical methodology, work which has been responded to and criticized by philosophers and historians of mathematics more on its own terms, a significant body of writing in the 30 years since its appearance has used it as a pertinent cognate text appropriable for school mathematics education. In this paper, we contrast the responses these two fields have generated, with an emphasis on that of mathematics education. Doing so offers a potentially salutary case study of how challenging and fraught it can be at times to undertake work at the nexus of history and philosophy on the one hand, while at the same time seeking to explore its possible relevance and significance for education. As our title suggests, we are concerned about the proliferating Lakatos personas that seem to exist, including a growing range of self-styled reform or progressive educational practices which get attributed to him.     

Metaphors of hierarchy in mathematics education discourse: the narrow path

This paper adopts a rhetorical perspective in order to examine language about children in the discourse of mathematics education through a study of metaphor. Previous research has tended to emphasize the notion of ‘beliefs’, which locates responsibility for problematic conceptions of children within the heads of individuals, particularly practising and preservice teachers. Using the notion of metaphor, this paper examines several texts in US mathematics education, including conversations in an elementary classroom, a university mathematics methods classroom, mathematics textbooks, and standards documents. All of these texts draw on the metaphor of children’s learning as travel along a physical path, which supports talking and thinking about children in hierarchical ways. The dominance of this metaphor presents a new challenge for teacher educators concerned with equity: that of examining their own language and practices for hierarchical language.     

Teachers’ conceptions of representation in middle school mathematics

Current reform efforts call for an emphasis on the use of representation in the mathematics classroom across levels and topics. The aim of the study was to examine teachers’ conceptions of representation as a process in doing mathematics, and their perspectives on the role of representations in the teaching and learning of mathematics at the middle-school level. Interviews with middle school mathematics teachers suggest that teachers use representations in varied ways in their own mathematical work and have developed working definitions of the term primarily as a product in problem solving. However, teachers’ conception of representation as a process and a mathematical practice appears to be less developed, and, as a result, representations may have a peripheral role in their instruction as well. Further, the data suggested that representation is viewed as a topic of study rather than as a general process, and as a goal for the learning of only a minority of the students—the high-performing ones. Implications for mathematics teacher education, prospective and practicing, are discussed.     

萨特关于匮乏的思想及其与马克思的差异

关于匮乏的思想是萨特人学理论的基石。在萨特看来,整个人类的发展都是在同匮乏作艰苦的斗争;匮乏是我们历史的根本关系,是对我们同物质单一关系的偶然确定;阶级的对立是匮乏的必然结果;匮乏已经内在化为人的非人性的人性结构。在对匮乏问题的看法上,萨特与马克思有根本差异。萨特还认为,匮乏是人类历史发展可能性的基础,但不是实在性的基础;劳动既是匮乏的实现也是对它的超越,由此形成我们的历史。     

Institutions influencing mathematics students' private work: A factor of academic achievement

The aim of the research presented in this paper is to contribute to our knowledge about problem solving in mathematics. My purpose in this paper is to compare, from this point of view, two very different institutions in the French tertiary education system, with the intention to interpret the chronic inequality of performance in problem solving between populations of mathematics students coming from these institutions. Problem solving knowledge and skills are not an explicit objective of teaching and their development depends largely on the student's private mathematical activity. This hypothesis is the reason why the inquiry aims at comparing mathematics students' ways of working as they study in both institutions. The results of the research are interpreted, on the institutional level, as effects of differences between the two teaching systems. This revised version was published online in July 2006 with corrections to the Cover Date.