Experiencing equivalence but organizing order

The notion of equivalence relation is arguably one of the most fundamental ideas of mathematics. Accordingly, it plays an important role in teaching mathematics at all levels, whether explicitly or implicitly. Our success in introducing this notion for its own sake or as a means to teach other mathematical concepts, however, depends largely on our own conceptions of it. This paper considers various conceptions of equivalence, in history, in mathematics today, and in mathematics education. It reveals critical differences in the notion of equivalence at different points in history and a meaning for equivalence proposed by mathematicians and mathematics educators that is at variance with the ways that learners may think. These differences call into question the most popular view of the subject: that the mathematical notion of equivalence relation is the result of spelling out our experience of equivalence. Moreover, the findings of this study suggest that the standard definition of an equivalence relation is ill-chosen from a pedagogical point of view but well-crafted from a mathematical point of view.     

数学哲学与数学方法论研究的若干课题

在数学研究中,在数学哲学和方法论研究中,在数学教育中蕴藏着大量的问题与课题.包括:希尔伯特、徐利治、波利亚等数学家和数学教育家的问题;教材编写、考试命题、数学共同体、数学辩证法以及复杂性研究等问题.     

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.     

Recignising commonalities and reconciling differences in mathematics education

Data from a study of the practices of research mathematicians are used to highlight an exploration of commonalities across and differences between learners in mathematics classrooms. Such commonalities and differences are, it is claimed, central to a teacher's understandings of how to address her role and responsibilities. Commonalities discussed are those between learners,across mathematics and between communities of practice. Differences between syllabus and practice, between school mathematics,mathematics in the world, and academic mathematics and, finally, between members of the different communities of students,teachers of mathematics, mathematics educators, and mathematicians are explored. This revised version was published online in July 2006 with corrections to the Cover Date.     

Mathematical experiences and parental involvement of parents who are and who are not mathematicians

Previous studies suggest that parental involvement in children’s mathematics education is more established for parents who feel competent in mathematics. This qualitative study aimed to gain an in-depth insight into the experiences of parental involvement of two different groups of parents: those who are mathematicians and those who are not. Data were collected through narrative interviews with parents. A thematic analysis of the data revealed findings within two distinct but interrelated themes: parents’ mathematical experiences and parental involvement in their children’s mathematics education. The findings indicated that the two groups of participating parents differ in their own experiences of mathematics as well as in their parental involvement. The main difference in parental involvement was indicated in the area of children’s school mathematics, since mathematician parents, compared to non-mathematician parents, according to their narratives almost never get involved in their children’s mathematics homework. In addition, the data revealed a large gap in the coverage and content of the mathematical activities that parents in both groups provided to their children.     

Imagining the mathematician: young people talking about popular representations of maths

This paper makes both a critical analysis of some popular cultural texts about mathematics and mathematicians, and explores the ways in which young people deploy the discourses produced in these texts. We argue that there are particular (and sometimes contradictory) meanings and discourses about mathematics that circulate in popular culture, that young people use these as resources in identity making as (non-)mathematicians, negotiating their meaning in ways that are not always predictable, and that their influence on young people is diffuse and nevertheless important. The paper discusses the discourses that prevail in some of the popular cultural images of mathematics and mathematicians that came up in our research. We show how mathematics is represented as a secret language, while mathematicians are often mad, mostly male and almost invariably white. We then explore how young people negotiate these discourses, positioning themselves in relation to mathematics. Here we draw attention to the fact that both those who continue with mathematics after it ceases to be compulsory and those who do not, deploy similar images of mathematics and mathematicians. What is different is how they respond to and negotiate these images.     

李善兰的数学哲学思想——纪念李善兰诞生200周年

李善兰(1811-1882)是19世纪我国最重要的数学家,他的数学成就在我国近代科学史上具有十分独特的地位。他的数学成就与他的数学哲学思想是分不开的。他的数学哲学思想主要之点是:数学是富国强兵的工具;研究数学的方法和步骤是"深思"、"通其法"和"解明之";要继承,更要创新;会通中西数学;以研究数学为主,会通相关学科。     

How Mathematicians Obtain Conviction: Implications for Mathematics Instruction and Research on Epistemic Cognition

The received view of mathematical practice is that mathematicians gain certainty in mathematical assertions by deductive evidence rather than empirical or authoritarian evidence. This assumption has influenced mathematics instruction where students are expected to justify assertions with deductive arguments rather than by checking the assertion with specific examples or appealing to authorities. In this article, we argue that the received view about mathematical practice is too simplistic; some mathematicians sometimes gain high levels of conviction with empirical or authoritarian evidence and sometimes do not gain full conviction from the proofs that they read. We discuss what implications this might have, both for mathematics instruction and theories of epistemic cognition.     

论中国现代数学的早期奠基历程

中国现代数学的奠基是在落后于西方数学200年的基础上,经过先辈数学家们走出国门,艰苦求学,然后回国创办高等教育,发展中国的现代数学.中国现代数学的奠基历程经历了从20世纪初到1929年间30年的艰难起步,回顾前辈数学家们开拓中国现代数学的这段艰苦历程,感受他们高度的民族自强精神和卓越的科学创造才能.     

青浦经验:数学活动经验的教育表达

数学的本质是有一套提出问题、解决问题的思想和方法.人们的数学活动经验是在做数学的过程形成的.真正的数学教学理论的本质是数学活动经验的教育表达.青浦经验的4条原理(情意原理、序进原理、活动原理、反馈原理)在教学实践中经受了时间检验的原因是从教育的侧面反映了数学知识的发生、发展和应用的规律.     

Practice and conceptions: communicating mathematics in the workplace

The study examined the experience of communication in the workplace for mathematics graduates with a view to enriching university curriculum. I broaden the work of Burton and Morgan (2000), who investigated the discourse practices of academic mathematicians to examine the discourse used by new mathematics graduates in industry and their perceptions of how they acquired these skills. I describe the different levels of perception of discourse needs and of how they gained the necessary skills. At the lowest level, they learnt through trying out different approaches. At the next level, they were assisted by colleagues or outside situations. At the highest level, a small group viewed communication and interpersonal skills as a scientific process and stood back and used their “mathematical” observation skills to model their behaviour. These graduates did not appear to have systematically studied communication as part of their degree and they were unaware of the power of language choices in the workplace. Those who were working as mathematicians had to come to grips with explaining mathematical concepts to a wide range of people with varying mathematical skills but who generally were considerably less skilled in mathematics. The study revealed that these graduates were seriously underprepared in many aspects for joining the workforce. Many found it difficult to adapt to dealing with colleagues and managers, and developing communication skills was often a matter of trial and error.     

Seeking Diversity in Mathematics Education: Mathematical Modeling in the Practice of Biologists and Mathematicians

Although reform efforts in mathematics education have called for more diverse views of mathematics, there have been few studies of how mathematics is used and takes form in practices outside of mathematics itself. Thus legitimate diverse models have largely been missing in education. This study attempts to broaden our understanding of mathematics by investigating how applied mathematicians and biologists, working together to construct dynamic population models, understand these models within the framework of their perspective practices, that is how these models take on a role as 'boundary objects' between the two practices. By coming to understand how these models function within the practice of biology, the paper suggests that mathematics educators have the opportunity both to reevaluate their own assumptions about modeling and to build an understanding of the dialectic process necessary for these models to develop an epistemological basis that is shared across practices. Investigating this dialectic process is both important and missing in most mathematical classrooms.1     

关于数学教育若干问题与现象的忧与思——兼论数学教育的学科建设

对当下实践中的数学教育的问题与现象的深层思考亟待加强.在各种数学教育行为中,我们不仅要关注其正面的效果,还要考虑其可能的负面效应.数学教育工作者需要常常自问的是,自己的数学教育观念和行为是否适当?为了创建坚实有效的数学教育实践观,构建适合中国国情的数学教育学科体系已成为当务之急.     

巧用数学史厘清数学概念——以“小数的初步认识”为例

学习中恰当地融入数学史料知识,学生了解知识的产生、发展过程,才能使学生更好地达到博古通今。问渠那得清如许?为有源头活水来,数学史知识会成为学生学习的源头活水,数学家的故事可以激励学生不断奋进、前行。     

论数学观教育与研究

数学现教育是使学生对数学本身及与它有关的因素有所认识的教育。有意识的、自觉的数学观教育是实现数学观教育目标的关键所在。深入地、多方面地开展数学观教育研究是确立我国明确的数学教育理念的重要基础。     

论课程形态的数学文化及其教育价值的实现

数学文化的研究越来越受到关注与重视.作为课程形态的数学文化有其独特的定义及特征.其教育价值包括科学方面的教育价值和人文方面的教育价值,其实现应从树立突出文化特征的数学观、数学课程观、数学教学观开始;同时,还要注意两个方面的问题:(1)要根据实际情况有计划、有步骤渐进式地推行数学的人文方面的教育;(2)不要全盘否定以往的数学教育的价值.     

`A historical angle’, a survey of recent literature on the use and value of history in geometrical education

In this article we give a survey of recent literature on the use and value of the history of geometry in mathematics education. It turns out that many authors have contributed to the debate why we should apply history. Their arguments can be divided into conceptual, (multi-) cultural and motivational ones. The number of authors who concentrated on the methodological question “How could we introduce history into mathematics lessons?” is considerably smaller. Those who want to make history an integral part of mathematics education have to bridge the gap between theoretical arguments and practical ideas, and, connected with this,between historians and teachers, and ultimately between mathematicians from the past and present day students. We divided the various surveyed publications, mainly journal articles, into categories according to a framework in order to clarify the discussion on the role of the history of geometry in education. Details about content and purpose of the articles are presented in two appendices, which make the articles more accessible both for further research and for practical teacher purposes.     

祖冲之与刘徽在国内外影响之比较

中国古代有许多优秀的数学家，其中祖冲之与刘微对现代数学教育的影响颇大，从数学理论的贡献看，应是刘徽大于祖冲之，但从国内外的影响看，他们是不同的，祖冲之在国内的影响要大于刘徽，在国际数学界他略逊于刘徽。     

HPM的历史渊源

数学史对数学教育的意义在19世纪已经引起西方数学史家的注意，法国数学家泰尔凯、英国数学家德摩根等都是其中重要的先驱者．20世纪上叶，一些欧美数学家，如卡约黎、庞加莱、史密斯等大力提倡数学史在数学教学中的运用．20世纪70年代，HPM成为一个独立的学术研究领域．今天，HPM已成为国际数学教育的新思潮之一．历史考察表明：HPM的学术源流是清晰可寻的。     

从中国三角学早期会通看科学传播中的创新点

西方三角学知识初入中国,偏重应用和计算而缺乏论证,清初数学家接受、学习的同时,在补充证明、精简公式、提高精度以及重构西方三角学上取得了创新成果,他们以会通中西为目的,以探寻数学内在原理、融合中西数学为己任,彰显出一股求真务实的科学精神。