数学史融入数学教育的有效途径与实施建议

数学史研究数学概念、数学思想与数学方法的起源与发展,可以毫不夸张地说,数学的历史就是数学思想方法的发展史.在五千年的数学历史长河中,伴随着每次重大的数学发现,例如数学史上的三次危机、解析思想以及非欧几何,都有重要的数学思想方法诞生.因此数学思想方法是数学史的灵魂与精髓,而数学教育的最终目的是为了培养学生的数学思维和数学能力,数学思维与能力主要是通过掌握数学知识、灵活运用数学思想方法形成的.因此在教学中,提炼数学史中的思想方法使之为数学教育服务是数学教育的一个当务之急.     

初中数学发生教学法的策略与应用——以北师大版“字母表示数”为例

德国生物学家海克尔在1866年提出了"生物发生原理",即"个体发育史重蹈种族发展史".将此类推于数学教育将得出:个体对数学知识的理解过程遵循数学知识的发生、发展过程.把数学史作为教学线索,不明确地谈论数学史,用数学史来启示教学,这就是数学发生教学法.一、发生教学法的策略运用发生教学法进行教学设计的关键在于教师,对教师的要求是:(1)要全面了解所教主题的历史;(2)要理解该主题历史发展过程中的关键环节;(3)掌握一个环     

新教材应增加数学史、数学家和数学新发展的简介的内容

新编高中数学教学大纲 (供实验用 )已于1 996年制定出来 .供实验用的教材也已经编写出来 .在修订新教材时 ,我希望能够增加数学史 ,数学家和现代数学知识简介的内容 .与物理 ,化学教材相比 ,我们的数学教材一直缺少对数学史 ,数学家和现代数学知识简介的内容 .我们的学生学完小学、初中、高中的课程 ,却连几个著名的数学家也不知道 ,不知道数学的悠久、辉煌而曲折的发展史 ,也不了解现代数学最新的发展情况 ,我认为这是很大的欠缺 .应该让学生知道一点数学史的知识 .通过学习数学史 ,可以培养学生良好的数学品质 ,激发学生对数学的兴趣 ,提…     

把静态的知识转化为动态的问题

《全日制义务教育数学课程标准 (实验稿 )》明确提出 ,要对数学教材、学生的数学学习方式、教师的课堂教学与评价进行全面改革 .为了实现课程改革目标 ,我们有必要深入研究如何在教材编写和教学中“把静态的知识转化为动态的问题” .1　数学知识与数学问题1 1　数学知识从《数学课程标准》“总体目标”中的阐述可以看出 :数学知识包括数学事实和数学活动经验 .这说明数学知识不仅包括“客观性知识” ,即那些不因地域和学习者而改变的基本概念、数学命题(公理、定理、公式、定律、法则 )、数学方法、数学史知识等 ,而且包括从属于学生自己的…     

对数学史在数学教学中作用的思考

1 问题的提出 目前,在全世界最普遍开设的教育课程是数学,其开设时间之长,惟有本国语言文学课程可以与之相比[1],它是学生必须学习的主要课程.一个人从小学、中学到大学所学到的数学知识,犹如大海中的一碗水.在步人工作岗位后如果不是在与数学相关的领域工作,所学过的具体的数学定理、公式和解题方法可能大多用不上了,而且还会遗忘.然而不管从事何种工作,铭刻在头脑中的数学素养即数学思想、推理方法、研究方法和求知能力,将伴随其终身,促使其去不断地探索新的知识.而数学史,实际上就是一部数学思想史,如何将数学史恰当地融入到数学教学中,以帮助学生加深对数学的理解,培养学生的创造性思维能力和激发学习数学的兴趣,这正是我们在数学教学中要思考的问题.     

圆的面积:从历史到课堂

关于数学史对数学教学的价值,学术界已经有过广泛而深入的讨论.从知识目标上说,数学史帮助学生理解数学;从过程与方法目标上说,数学史提供了丰富的问题解决方法,可以拓宽学生的思维;从情感、态度和价值观的目标上说,数学史增加学生的学习兴趣、激发学生的学习动机,使数学变得更亲和、更令人愉悦、更激动人心,揭示出数学作为人类文化活动的本质.这些价值在数学课堂上究竟能否实现?笔者在沪太路教育发展区项目“HPM与初中数学教师专业发展”的引领下,在六年级上学期“圆的面积”的教学中融入数学史,通过课堂观察、问卷调查和访谈,收集学生的反馈信息,并结合专家和听课教师的反应,得出结论,从而对上述问题作出回答.     

数学文化渗透数学课堂教学的探索

高中即将实施的新课标中明确提出要将数学文化融入课程内容之中,课堂是课程的落脚点,数学文化融入课程首先要让数学文化走进课堂.数学文化是以数学学科为核心,不仅包括数学知识、方法、思想,还包括数学精神、数学史、数学审美与数学应用     

重视数学史在数学教育中的作用

为了全面了解数学科学 ,探索数学发展的规律 ,为了数学教育的目的 ,都应该开展数学史的教学与研究 ,进一步认识数学史在数学教育中的地位和价值 ,充分发挥数学史知识在进行素质教育方面的重要作用 .1　数学史在数学教育中的地位数学史是学习数学、认识数学的工具 .人们要弄清数学概念、数学思想和方法的发展过程 ,增长对数学的通识 ,建立数学的整体意识 ,就必须运用数学史作为补充和指导 .特别是 ,现代数学的体系犹如“茂密繁盛的森林” ,使人“站在外面窥不见它的全貌 ,深入内部又可能陷身迷津” ,数学史的作用就是指引方向的“路标” ,给…     

数学文化渗透数学课堂教学的探索

高中即将实施的新课标中明确提出要将数学文化融入课程内容之中,课堂是课程的落脚点,数学文化融入课程首先要让数学文化走进课堂.数学文化是以数学学科为核心,不仅包括数学知识、方法、思想,还包括数学精神、数学史、数学审美与数学应用价值,教师不可将数学文化狭隘成数学史.     

来稿须知

本刊衷心欢迎以下投稿 :1 以高观点、新方法、新角度讲解数学知识的文章 ;介绍适合于教师及学生的有关数学新知识的内容 ;2 有关新课标、新课程讲解、争鸣、反思、调研及解读的文章 ;3 有关数学教育理论、数学研究、教法研究、教材研究、优秀课例、解题教学方面的文章 ;4 高考研究总结与复习方面的文章 ,内容切实有助于学生提高数学能力 ,掌握数学知识 ,理解数学思想及方法 ;5 初等数学研究、数学竞赛方面的文章 ,应以有助于提高对数学本质的认识 ,注重数学知识的关系 ;6 有关国外数学、数学家、数学史、数学与文化等方面的文章 ,注重趣…     

Polynomial calculus: rethinking the role of calculus in high schools

Access to advanced study in mathematics, in general, and to calculus, in particular, depends in part on the conceptual architecture of these knowledge domains. In this paper, we outline an alternative conceptual architecture for elementary calculus. Our general strategy is to separate basic concepts from the particular advanced techniques used in their definition and exposition. We develop the beginning concepts of differential and integral calculus using only concepts and skills found in secondary algebra and geometry. It is our underlining objective to strengthen students' knowledge of these topics in an effort to prepare them for advanced mathematics study. The purpose of this reconstruction is not to alter the teaching of limit-based calculus but rather to affect students' learning and understanding of mathematics in general by introducing key concepts during secondary mathematics courses. This approach holds the promise of strengthening more students' understanding of limit-based calculus and enhancing their potential for success in post-secondary mathematics.     

高等数学教学方法的研究与实践

结合自身的教学经验,总结归纳了高等数学的教学方法与实践过程.主要从以下几个方面作了阐述:如何激发学生学习高等数学的兴趣,变被动学习为主动学习;如何将抽象问题直观化;如何将归纳和类比的方法应用到高等数学教学中;如何在教学中拓宽知识面,培养学生的发散思维.     

Conception of expert mathematics teacher: a comparative study between Hong Kong and Chongqing

This paper reports the similarities and differences in how “expert mathematics teacher” is conceptualized by mathematics educators in Hong Kong and Chongqing, two cities in China which share similar but different cultural and social backgrounds. Thirty-seven mathematics education researchers, school principals with mathematics education background, and mathematics teachers were interviewed on their perceptions of expert mathematics teacher. It is found that in both cities an expert mathematics teacher should have a profound knowledge base in mathematics, teaching, and students; strong ability in teaching; and a noble personality and a spirit of life-long learning. As for differences, an expert mathematics teacher should have the ability to conduct research, mentor other teachers, and have profound knowledge of examination and educational theories in Chongqing. These attributes were not found in Hong Kong. These similarities and differences are discussed, and relevant social and cultural factors in the two contexts are examined.     

A case study of pedagogy of mathematics support tutors without a background in mathematics education

This study investigates the pedagogical skills and knowledge of three tertiary-level mathematics support tutors in a large group classroom setting. This is achieved through the use of video analysis and a theoretical framework comprising Rowland's Knowledge Quartet and general pedagogical knowledge. The study reports on the findings in relation to these tutors’ provision of mathematics support to first and second year undergraduate engineering students and second year undergraduate science students. It was found that tutors are lacking in various pedagogical skills which are needed for high-quality learning amongst service mathematics students (e.g. engineering/science/technology students), a demographic which have low levels of mathematics upon entering university. Tutors teach their support classes in a very fast didactic way with minimal opportunities for students to ask questions or to attempt problems. It was also found that this teaching method is even more so exaggerated in mandatory departmental mathematics tutorials that students take as part of their mathematics studies at tertiary level. The implications of the findings on mathematics tutor training at tertiary level are also discussed.     

Knowledge and confidence of pre-service mathematics teachers: the case of fraction division

To make teacher preparation and professional development effective, it is important to find out possible deficiencies in teachers’ knowledge as well as teachers’ own perceptions about their needs. By focusing on pre-service teachers’ knowledge of fraction division in this article, we conceptualize the notion of pre-service teachers’ knowledge in mathematics and pedagogy for teaching as containing both teachers’ perceptions of their preparation and their mathematics knowledge needed for teaching. With specific assessment instruments developed for pre-service middle school teachers, we focus on both pre-service teachers’ own perceptions about their knowledge preparation and the extent of their mathematics knowledge on the topic of fraction division. The results reveal a wide gap between sampled pre-service middle school teachers’ general perceptions/confidence and their limited mathematics knowledge needed for teaching fraction division conceptually. The results suggest that these pre-service teachers need to develop a sound and deep understanding of mathematics knowledge for teaching in order to build their confidence for classroom instruction. The study’s findings indicate the feasibility and importance of conceptualizing the notion of teachers’ knowledge in mathematics and pedagogy for teaching to include teachers’ perceptions. The applicability and implications of this expanded notion of teachers’ knowledge is then discussed.     

Mathematics university teachers' perception of pedagogical content knowledge (PCK)

Teaching mathematics in university levels is one of the most important fields of research in the area of mathematics education. Nevertheless, there is little information about teaching knowledge of mathematics university teachers. Pedagogical content knowledge (PCK) provides a suitable framework to study knowledge of teachers. The purpose of this paper is to make explicit the perception of mathematics university teachers about PCK. For this purpose, a phenomenological study was done. Data resources included semi-structured interviews with 10 mathematics university teachers who were in different places of the mathematics university teaching experience spectrum. Data analysis indicated a model consisting of four cognitive themes which are mathematics syntactic knowledge, knowledge about mathematics curriculum planning, knowledge about students' mathematics learning and knowledge about creating an influential mathematics teaching–learning environment. Besides, it was found out that three contextual themes influenced on PCK for teaching mathematics in university levels which were the nature of mathematics subjects, university teachers' features and terms of learning environment.     

Potential scenarios for Internet use in the mathematics classroom

Research on the influence of multiple representations in mathematics education gained new momentum when personal computers and software started to become available in the mid-1980s. It became much easier for students who were not fond of algebraic representations to work with concepts such as function using graphs or tables. Research on how students use such software showed that they shaped the tools to their own needs, resulting in an intershaping relationship in which tools shape the way students know at the same time the students shape the tools and influence the design of the next generation of tools. This kind of research led to the theoretical perspective presented in this paper: knowledge is constructed by collectives of humans-with-media. In this paper, I will discuss how media have shaped the notions of problem and knowledge, and a parallel will be developed between the way that software has brought new possibilities to mathematics education and the changes that the Internet may bring to mathematics education. This paper is, therefore, a discussion about the future of mathematics education. Potential scenarios for the future of mathematics education, if the Internet becomes accepted in the classroom, will be discussed.     

Chinese elementary mathematics teachers’ knowledge in mathematics and pedagogy for teaching: the case of fraction division

In this study, we investigated the extent of knowledge in mathematics and pedagogy that Chinese practicing elementary mathematics teachers have and what changes teaching experience may bring to their knowledge. With a sample of 18 mathematics teachers from two elementary schools, we focused on both practicing teachers’ beliefs and perceptions about their own knowledge in mathematics and pedagogy and the extent of their knowledge on the topic of fraction division. The results revealed a gap between these teachers’ limited knowledge about the curriculum they teach and their solid mathematics knowledge for teaching, as an example, fraction division. Moreover, senior teachers used more diverse strategies that are concrete in nature than junior teachers in providing procedural justifications. The results suggested that Chinese practicing teachers benefit from teaching and in-service professional development for the improvement of their mathematics knowledge for teaching but not their knowledge about mathematics curriculum.     

Mathematics teaching expertise development approaches and practices: similarities and differences between Western and Eastern countries

This paper comments on the other papers in this issue related to how “mathematics teaching expertise” is conceptualized and the approaches employed to facilitate its development in Western and Eastern countries. Similarities and differences are found to exist in the conceptualization of mathematics teaching expertise and the development approaches employed. The papers in this issue share the similarity of exploring mathematics teaching expertise from the perspective of knowledge. Under the influence of this perspective, the approaches mentioned in the papers mainly focus on the development of teachers’ knowledge. A feature in common among teacher development approaches employed in Western countries is to let teachers attend some courses or training programs designed or organized by mathematics teacher educators at universities. In contrast, teacher development approaches employed in Eastern countries, particularly those employed in Mainland China, are relatively more practical in nature and directly related to teachers’ needs, like learning from observing exemplary teaching. This shows that the conception of mathematics teaching expertise and development approaches are culturally and contextually dependent. It is argued that a broader perspective of mathematics teaching expertise should be taken to explore mathematics teaching expertise and its development, and teacher expertise development should be conceptualized as a complex system rather than as some separated knowledge, skills and techniques.     

Assessment of teacher knowledge across countries: a review of the state of research

This review presents an overview of research on the assessment of mathematics teachers’ knowledge as one of the most important parameters of the quality of mathematics teaching in school. Its focus is on comparative and international studies that allow for analyzing the cultural dimensions of teacher knowledge. First, important conceptual frameworks underlying comparative studies of mathematics teachers’ knowledge are summarized. Then, key instruments designed to assess the content knowledge and pedagogical content knowledge of future and practicing mathematics teachers in different countries are described. Core results from comparative and international studies are documented, including what we know about factors influencing the development of teacher knowledge and how the knowledge is related to teacher performance and student achievement. Finally, we discuss the challenges connected to cross-country assessments of teacher knowledge and we point to future research prospects.