The Notion of Proof in the Context of Elementary School Mathematics

Despite increased appreciation of the role of proof in students’ mathematical experiences across all grades, little research has focused on the issue of understanding and characterizing the notion of proof at the elementary school level. This paper takes a step toward addressing this limitation, by examining the characteristics of four major features of any given argument – foundation, formulation, representation, and social dimension – so that the argument could count as proof at the elementary school level. My examination is situated in an episode from a third-grade class, which presents a student’s argument that could potentially count as proof. In order to examine the extent to which this argument could count as proof (given its four major elements), I develop and use a theoretical framework that is comprised of two principles for conceptualizing the notion of proof in school mathematics: (1) The intellectual-honesty principle, which states that the notion of proof in school mathematics should be conceptualized so that it is, at once, honest to mathematics as a discipline and honoring of students as mathematical learners; and (2) The continuum principle, which states that there should be continuity in how the notion of proof is conceptualized in different grade levels so that students’ experiences with proof in school have coherence. The two principles offer the basis for certain judgments about whether the particular argument in the episode could count as proof. Also, they support more broadly ideas for a possible conceptualization of the notion of proof in the elementary grades.     

我国民事诉讼举证责任分担原则探析

鉴于我国民事诉讼法第６４条关于举证责任制度的规定较为简单，无法适应诉讼实践中千变万化的客观情势，因而有进一步修改和完善之必要，为此，笔者提出了一个公正、合理的举证责任分担的标准。     

量刑阶段的证明责任与证明标准问题研究--从新常见罪量刑意见展开

量刑与定罪有着不同的司法功能,在量刑阶段,寻求设计与之相适应的证明责任、证明标准以及证据规则等制度内容,是我国定罪量刑程序分离改革过程中的重要组成部分。在深化司法改革大背景下,结合新刑事诉讼法增加“排除合理怀疑”的证明基本要求、扩宽当事人参与法庭调查与辩论范围、建立量刑建议制度等内容,可以得出：在量刑阶段建立相对独立的证明标准、证明责任等制度,既有其必要性,也有其可行性。     

微分在不等式证明中的应用

利用函数的微分证明不等式的思想方法,在诸多数学分析论著中有所提及,是微分的一个重要应用。其主要方法有:利用函数的单调性证明不等式;利用函数的凸凹性证明不等式;利用Lagrange微分中值定理或泰勒公式证明不等式;利用求函数极值的方法证明不等式。     

Deficit or Difference? the Role of Students' Epistemologies of Mathematics in their Interactions with Proof

The ability to handle proof is the focus of a number of well-documented complaints regarding students' difficulties in encountering degree-level mathematics. However, in addition to observing that proof is currently marginalised in the UK pre-university mathematics curriculum with a consequent skills deficit for the new undergraduate mathematics student, we need to look more closely at the nature of the gap between expert practice and the student experience in order to gain a full explanation. The paper presents a discussion of first-year undergraduate students' personal epistemologies of mathematics and mathematics learning with illustrative examples from 12 student interviews. Their perceptions of the mathematics community of practice and their own position in it with respect to its values, assumptions and norms support the view that undergraduate interactions with proof are more completely understood as a function of institutional practices which foreground particular epistemological frameworks while obscuring others. It is argued that enabling students to access the academic proof procedure in the transition from pre-university to undergraduate mathematics is a question of fostering an epistemic fluency which allows them to recognise and engage in the process of creating and validating mathematical knowledge.     

Connecting Science and Mathematics: The Nature of Proof and Disproof in Science and Mathematics

Disagreements exist among textbook authors, curriculum developers, and even among science and mathematics educators/researchers regarding the meanings and roles of several key nature-of-science (NOS) and nature-of-mathematics (NOM) terms such as proof, disproof, hypotheses, predictions, theories, laws, conjectures, axioms, theorems, and postulates. To assess the extent to which these disagreements may exist among high school science and mathematics teachers, a 14-item survey of the meanings and roles of the above terms was constructed and administered to a sample of science and mathematics teachers. As expected, the science teachers performed better than the mathematics teachers on the NOS items (44.1 versus 24.7%, respectively) and the mathematics teachers performed better than the science teachers on the NOM items (59.0 versus 26.1%, respectively). Nevertheless, responses indicated considerable disagreement and/or lack of understanding among both groups of teachers concerning the meanings/roles of proof and disproof and several other key terms. Therefore it appears that these teachers are poorly equipped to help students gain understanding of these key terms. Classroom use of the If/and/then/Therefore pattern of argumentation, which is employed in this paper to explicate the hypothesis/conjecture testing process, might be a first step toward rectifying this situation.     

Investigating the Guidance Offered to Teachers in Curriculum Materials: The Case of Proof in Mathematics

Despite widespread agreement that proof should be central to all students’ mathematical experiences, many students demonstrate poor ability with it. The curriculum can play an important role in enhancing students’ proof capabilities: teachers’ decisions about what to implement in their classrooms, and how to implement it, are mediated through the curriculum materials they use. Yet, little research has focused on how proof is promoted in mathematics curriculum materials and, more specifically, on the guidance that curriculum materials offer to teachers to enact the proof opportunities designed in the curriculum. This paper presents an analytic approach that can be used in the examination of the guidance curriculum materials offer to teachers to implement in their classrooms the proof opportunities designed in the curriculum. Also, it presents findings obtained from application of this approach to an analysis of a popular US reform-based mathematics curriculum. Implications for curriculum design and research are discussed.     

民事诉讼举证责任初探

举证责任是民事诉讼制度的重要组成部分,其性质应是法定的当事人应履行的诉讼义务.针对我国目前民事举证责任制度存在的不足,我们建议将现行的较分散的规定整和到法律中去,并且充分运用司法解释这一有效的工具以避免实体法的滞后.     

确界原理的一个简单证明

用反证法给出了确界原理的一个简单证明。     

刍议治安行政案件的证明标准

诉讼证明标准历来是诉讼法学界研究证据问题的重点,近年来刑事诉讼的证明标准发生了由“客观真实”到“法律真实”的转变,衍生了多元化证明标准理论,这也是治安行政案件的证明标准的理论基础;多层次证明标准理论主张根据案件和证据的性质特征采取不同的证明标准,在治安行政案件中实行多层次的证明标准有着重要的理论基础和实践价值。     

浅析票据法上的拒绝证明

拒绝证明是一个抽象的概念,其外在形式具有多样性.在持票人直接取得拒绝证明困难时,一些机关的文件也起着拒绝证明的替代作用.在法定的条件下,持票人可以免除提供拒绝证明而行使追索权.     

中学数学教学方法对本科数学专业教学的启示

当前大学数学专业专业课程的教学仍以灌输式为主,而中学数学教改中涌现的各种新方法重视教师和学生的相互合作,能调动学生学习的主动性,有利于培养学生的能力.这对数学专业专业课程的教学有重要的启示作用.     

A Critical Examination of Three Factors in the Decline of Proof

Proof seems to have been losing ground in the secondary mathematics curriculum despite its importance in mathematical theory and practice. The present paper critically examines three specific factors that have lent impetus to the decline of proof in the curriculum: a) The idea that proof need be taught only to those students who intend to pursue post-secondary education, b) the view that deductive proof need no longer be taught because heuristic techniques are more useful than proof in developing skills in reasoning and justification, c) the idea that deductive proof might profitably be abandoned in the classroom in favour of a dynamic visual approach to mathematical justification. The paper concludes that proof should be an essential component in mathematics education at all levels and compatible with both heuristic techniques and dynamic visual approaches.     

体现数学评价多元化的重要方式——数学日记述评

数学评价改革的核心是由单一的评价方式转向多元的评价方式。作为体现数学多元评价方式之一的数学日记,日益受到理论和实践的关注。本文阐述了数学日记在对于学生进行数学交流、数学思考、数学情感以及学生个性发展方面的价值;指出了基于数学课程三维目标下的数学日记内容以及如何在教学中实施并进行恰当地评价。     

Approaches to Multicultural Education in Preservice Teacher Education: Philosophical Frameworks and Models for Teaching

The authors in this article connect teacher education to multiple theoretical frameworks around the ongoing debate and issues of multicultural education. Connecting conservative, liberal, and radical theories of multicultural education particularly to preservice teachers, the authors argue that a more eclectic theoretical avenue must be striven and struggled for if there exists any hope in transforming schools, particularly, as they note, in urban environments. Practical avenues are discussed that promote such a multilayered interpretive/analytical approach to social change.     

线性空间理论中一个不能推广的证明

设V1，V2是数域F上的线性空间V的真子空间，很容易证明V1UV2真包含在V中．对于V中的m个真子空间V1，V2，…，Vm，易见mUi=1 V1也真包含在V中．但m=2时的证明方法不能推广到任意的m，否则，可以得到一个假命题．     

运用推广与引申的方法培养学生的创新能力

推广引申是数学研究与发展的重要方式，也是数学教学中培养不创新能力的良好途径。可以把推广引申的常见做法教给中学生，将推广引申结合到数学教学的各个方面使之成为学生建构知识的良好形式，并努力创造环境条件，激励学生以促进创新能力的培养。     

求诚务实：深化语文教学改革的根本途径

当前，语文教学中大量存在的形式主义倾向严重影响了新课标的落实。要克服这些弊端，深化语文教学改革，根本的途径就是语文教师和学生去伪求诚，去虚求实。只有教与学的双方都求诚务实，互相配合和协调，才能实现语文教学过程的优化。     

电磁场正则量子化方法的协变性证明

在本文中，利用电磁场正则量子化方法得出等时对易关系理论的不变性，在坐标系平移和空间旋转下予以证明．为了做到这些，除拉氏密度L(x)为标量外，还要导出势A^v(x)满足的务件，他们由M^ck生成的一个无穷小Lorentz变换得到，然后，证明Maxwell方程是Lorentz协变的，也证明了等时对易关系在变换后的坐标中仍然是正确的．     

巧求两小球对心弹性碰撞后的速度

本文用对称性原理推导出两小球对心弹性碰撞后的速度公式,具有求解简单,宜于推广,以高中生学习物理有很大的帮助。