数学应用：对应用题的超越——对小学数学教材中“数学应用”编写的几点思考

《全日制义务教育数学课程标准(实验稿)》取消了"应用题"这一内容体系,同时提出了"解决问题"的目标要求。面对这一调整,教材编写者和一线教师都面临着巨大的挑战,在实际操作中也出现了概念理解的混乱。实际上,我们应从更为广义的角度去理解数学的应用,将"应用题"的内容范围加以扩展,建立一个更为完善的"数学应用"体系。而在具体编写上,又应以"问题解决"的相关理论为指导,在"问题解决"的大背景下编写"数学应用"。"数学应用"是对"应用题"的一种超越,其内容覆盖面要更广,形式要更多样,要体现问题解决的一般过程,要体现解决策略的多样化,要注重提高学生提出问题的能力,要着眼于数学思维品质与数学精神的全面提高。     

Mathematics Learning Development: the Role of Long-Term Retrieval

This study assessed the relation between long-term memory retrieval and mathematics calculation and mathematics problem solving achievement among elementary, middle, and high school students in nationally representative sample of US students, when controlling for fluid and crystallized intelligence, short-term memory, and processing speed. As hypothesized, structural equation modeling comparing elementary school students and middle and high school students revealed that long-term retrieval skills became a better predictor of both mathematics calculation and mathematics problem solving as age and grade increased. Future research should focus on the effectiveness of interventions to improve long-term retrieval skills in general, and arithmetic facts retrieval and problem solving procedures in particular, at all grades, including high school.     

析中学数学教与学过程中的定势化倾向

论述了当前中学数学教学过程中,由教材和教学方式带来的定势思维的问题,并用数学教育哲学中的一些观点,对此进行了分析和思考．     

Instruction,repetition, discovery: restoring the historical educational role of practice

This conceptual paper considers what it would mean to take seriously Freudenthal's suggestion that mathematics should be taught like swimming. The general claim being made is that “direct instruction” and “discovery” are not opposite but complementary, linked by repetitive yet explorative practice. This claim is elaborated through an empirical case of martial arts instruction. That repetitive practice can nonetheless be a fountainhead of discovery is explained using Bernstein's notion of repetition-without-repetition. Finally, we attend to parallels in canonical mathematics practice. Implications are discussed, with a focus on reconceptualizing direct instruction, repetition, and discovery as complementary and synergistic.     

Getting the fish out of the water: Considering benefits and problems of doing research on teacher education at an international level

Structure and content of teacher education depend on a deeper rationale, which is a result of cultural boundaries. At the same time teaching is a cultural practice that differs across countries. Like the water in the fish's tank, such cultural givens are too often invisible as we debate research designs. In this article, we focus in particular on the understanding of three main components of teacher education: mathematics, mathematics pedagogy and general pedagogy, and on juxtaposing two extreme models: Germany and the US. It turns out that benefits and problems of international comparisons are closely related to each other.     

Conceptualising Resources as a Theme for Teacher Education

In this report, I examine resources and their use in school mathematics. I do so from the perspective of mathematics teacher education and with a view to the practice of school mathematics. I argue that the effectiveness of resources for mathematical learning lies in their use, that is, in the classroom teaching and learning context. The argument pivots on the concepts of school mathematics as a hybrid practice and on the transparency of resources in use. These concepts are elaborated by examples of resource use within an in-service teacher education research project in South Africa. I propose that mathematics teacher education needs to focus more attention on resources, on what they are and how they work as an extension of the teacher in school mathematics practice. In so doing, the report provides a language with which mathematics teacher educators and mathematics teachers can investigate teachers' use of resources to support mathematical learning in particular and diverse contexts. This revised version was published online in September 2006 with corrections to the Cover Date.     

The curricular importance of mathematics: a comparison of English and Hungarian teachers’ espoused beliefs

This paper reports an interview study of 45 English and 10 Hungarian teachers of mathematics. The semi‐structured interviews focused on the teachers’ professional life‐histories and invited them to discuss their beliefs about the necessary subject content for the teaching and learning of mathematics. Substantial differences emerged between the two cohorts, which accord with well‐defined national perspectives on education in general and mathematics education in particular. They reflect, at national rather than individual levels, the expectations of the curricular frameworks within which teachers operate. English teachers tended to view mathematics as applicable number and the means by which learners are prepared for a world beyond school. Hungarian teachers privileged mathematics as problem‐solving and logical thinking.     

Experiences Situating Mathematical Problem Solving at the Core of Early Childhood Classrooms

Our goal in this article is to discuss the importance of problems in early childhood education for the child’s development and engagement with the mathematics existing in childhood culture. Our assumption is that an important task for young children’s education is to create a democratic and critical environment, in which multiplicity of perspectives is celebrated, along with diversity of concepts and practices, with movement between imaginary and real worlds. In light of this, the goal of this article is to defend a perspective for curriculum and for the role of the mathematics educator, promoting the learning of mathematics through problem solving in early childhood years. In order to discuss and illustrate this perspective we describe the pedagogical practices of two teachers who teach 4- and 5-years-olds, who create for their students an environment rich in problem solving and investigations. In both classrooms, all children individually succeeded in sharing their unique solutions and new knowledge constructed as a result of their inquiries. The experience provides evidence that problem solving affords children the opportunity to raise conjectures, to discuss possibilities and to draw conclusions, even if partial ones, that are then vetted by the group as the authors share their solutions. In this way, the work with problem solving nurtures cooperative learning and promotes the exploration of a diversity of ideas.     

数学量的一般特征及其对数学教育教学的意义

数学研究的对象是什么?它的一般特征是怎样的?这是数学哲学本体论中,一项久议不决的疑案,因而给数学教育教学造成了不良影响.恩格斯认为,数学是研究量的科学,林夏水提出了量的层次性学说,厘清量的一般特征对数学教育教学有着重要意义.     

基于Flash3D技术的小学立体几何教学平台的设计与实现

空间观念的培养是小学数学课程的重要目标之一。当前的教学过程中缺少可以操作的可视化学习工具,难以满足立体几何的教学需求,根据这一问题设计并实现了小学立体几何教学平台。文章阐述了该平台的设计理念、主要功能和系统架构,引进了新兴的Flash3D技术,并对著名的Flash3D引擎——Alternativa3D的开发流程进行了详细介绍,基于该引擎对平台进行实现。平台具有逼真的三维场景、丰富的感性资源、友好的交互方式等特点,是新技术在教学中应用的一次有益尝试。     

国外数学问题提出能力影响因素的研究述评——基于学生自身的知识经验和观念系统等“变量”因素

在数学教学中,学生问题提出能力的发展不仅与教师的教学有关,还受到学生自身已有的观念系统与知识经验等"变量"因素的影响。论述数感、符号意识、空间观念、推理能力、问题意识、学习方式等6个学生"变量"在数学教育中的大致发展进程和主要概念解释,对国外有关问题提出学生"变量"的研究成果进行分析和述评,为中国问题提出能力的培养和教学提供了借鉴和思考的方向:关注不同内容领域的问题提出特点研究;加强学生自身主观因素对问题提出能力的影响研究。     

如何用自然辩证法原理指导数学研究

自然辩证法原理对数学的研究具有指导和启发意义 ,它指出了数学研究的源泉 ,数学研究的一般方法 ,数学选题应注意的社会条件 ,我们要使数学为自然、人、社会的协调发展服务 .     

Primary Mathematics Teacher Education in Greece: Reality and Vision

This paper focuses on two main issues concerning the mathematics education of prospective primary school teachers in Greece: the integration of mathematics and pedagogy and the relation of theory to practice. In particular, specific teaching approaches are discussed concerning the problem of integration both in mathematics and in mathematics education courses. The problem of "theory – practice" is examined through an analysis of the kind of teaching practice in which the prospective teachers are involved. Finally, the constraints that the mathematics educators face and the impact of their work on the professional life of the teachers in the future are discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

Plato, Pascal, and the dynamics of personal knowledge

Abstract educational practices are to be based on proven scientific knowledge, not least because the function science has to perform in human culture consists of unifying practical skills and general beliefs, the episteme and the techne (Amsterdamski, 1975, pp. 43–44). Now, modern societies first of all presuppose regular and standardized ways of organizing both our concepts and our institutions. The explanatory schemata resulting from this standardization tend to destroy individualism and enchantment. But mathematics education is in fact the only place in which to treat the human subject’s relationship with mathematics. And that is what mathematics education is all about: make the human subject grow intellectually and as a person by means of mathematics. At first sight, mathematics, in its formal guise, seems the opposite of philosophy, because philosophy constructs concepts (meanings), whereas mathematics deals with extensions of concepts (sets). We shall, however, turn this problem into an instrument, using the complementarity of intensions and extensions of theoretical terms as our main device for discussing the relationship between philosophy and mathematics education. The complementarity of the “how” and the “what” of our representations outlines, in fact, the terrain on which epistemology and education are to meet.     

美国中小学数学教育的现状及思考

美国中小学数学课程计划充分体现多样化、自主化的特色,尤其是高中会提供丰富的课程计划供不同学生选择.高中数学必修课程一般至少修3个学分的数学,包括代数1、几何、代数2等内容.美国高中数学选修课程很多.美国中小学数学教材呈现教材多样、目标明确、利于教学、资源丰富等特色.美国学校课堂教学重视多种学习方式、重视基础知识与基本技能教学、重视基本方法教学、重视问题解决教学、注重分层教育.。     

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.     

The Use of Two-dimensional Models in Social Science: An autocritical review

A two-dimensional model, made for identification of scenarios of future developments of education in general and teacher education in particular, is critically scrutinised in the article. The concepts of 'dimension' that belongs to mathematics of space, and 'opposite' that belongs to logic are elaborated on in relation to how notions of dimensionality and of opposite are dealt with in the model. It is claimed that the model does not deal with dimensions at all and that there are no opposites between the end points of the two 'dimensions' in the model. The model is yet useful from other points of view, namely as a heuristic model for evoking thoughts on the tensions, dynamisms, tendencies and counter tendencies in present education policies. Models in social science are not true or false but rather useful or less useful in all their imperfection for illustrating social phenomena and for inspiring new paths of thought.     

中国逻辑必然推理思想探赜

从结构上看,数学是问题和问题解的集合,逻辑是问题和答案之间的桥梁。具体到中国古代数学,其一般结构表现为：问、答、术（图）、草。＂术（图）＂阐述解题原理和步骤,＂草＂给出详细的解题过程,二者的功能和西方数学中逻辑的功能完全相同。中国数学中的＂术（图）＂和＂草＂,就是中国古代数学中的中国逻辑。中国数学是中国逻辑必然推理的研究对象,二者具有相同的源流。     

Promoting female achievement in the sciences: Research & implications

It is anticipated that by the year 2000 Canadian women will make up approximately 50% of the Canadian labour force. Despite this seemingly positive trend toward equitable gender-based participation in the labour force, females are extremely under represented in the scientific and technological fields (Statistics Canada, 1993). Females who are excluded or exclude themselves from the study of mathematics and science, limit career options and advancement opportunities in areas that drive and dominate social and economic trends. The underutilization of females in careers dependent upon science and mathematics expertise extends beyond the issue of individual actualization of potential, and has important consequences for society; significantly, as a threat to the economic prosperity of the nation. The key questions associated with this problem are: What are the factors which delimit and enhance female participation and achievement in the sciences? What can counsellors, educators and parents do to change this trend? Previous research has explored several dimensions, however, the greatest emphasis has been given to the particular barriers girls and women face. Relatively limited work has been given to factors associated with female success in the sciences. This paper reviews our current understanding of the problem, and describes a current research study that attempts to address some of the problems associated with previous theory and research in this area.     

Values in the mathematics classroom

Abstract

Values, moral values and democratic values are attracting the attention of education researchers in general and mathematics education researchers in particular. Little research has studied pre-service teachers’ perceptions of values in the classroom, their perceptions of the relationship between the different variables of values in the classroom, as well as their relationship with the democratic society. The present research attempts to do so. Twenty-two graduate pre-service teachers who participated in ‘New trends in mathematics education’ course discussed how to cultivated values in the mathematics classroom. Moreover, they answered survey questions related to the cultivation of values in this classroom. We used a combination of deductive and inductive content analysis to characterize the pre-service teachers’ texts. The research results indicate that the pre-service teachers perceived values as encouraging students’ activity in the mathematics classroom. In addition, the pre-service teachers perceived values as encouraging specific categories of values needed as skills for the citizen in a democratic society, as creativity, critical thinking and metacognition.