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.     

中学数学课应注重逻辑思维能力的培养

中学数学课培养学生逻辑思维能力是整个数学教学的目的之一，是数学教学的一个重要方面。通过数学教学培养学生自觉地掌握并运用逻辑规律进行思维的能力，也就是学生能明确地使用概念、恰当地下判断，合乎逻辑地进行推理的能力。为此教师在数学课的教学中，应把起妈的形式逻辑知识和辩证逻辑知识贯穿其中。以形式逻辑知识为主，兼顾一点辩证逻辑知识。通过逻辑思维的教学，使学生能深刻地提示概念、判断、推理的本质，真正提高学生的数学水平。     

弗雷格《概念文字》的构造及其意义

弗雷格在研究逻辑和算术的关系时深切地认识到了语言的缺陷,因此他借鉴了数学思想,引入了断定、函数和自变元等符号,在传统逻辑自然语言和算术形式语言的基础上构造出了纯思维的形式语言,在历史上第一次建立了一阶谓词演算系统,开创了现代逻辑的新纪元.     

Three concepts or one? Students’ understanding of basic limit concepts

In many mathematics curricula, the notion of limit is introduced three times: the limit of a sequence, the limit of a function at a point and the limit of a function at infinity. Despite the use of very similar symbols, few connections between these notions are made explicitly and few papers in the large literature on student understanding of limit connect them. This paper examines the nature of connections made by students exposed to this fragmented curriculum. The study adopted a phenomenographic approach and used card sorting and comparison tasks to expose students to symbols representing these different types of limit. The findings suggest that, while some students treat limit cases as separate, some can draw connections, but often do so in ways which are at odds with the formal mathematics. In particular, while there are occasional, implicit uses of neighbourhood notions, no student in the study appeared to possess a unifying organisational framework for all three basic uses of limit.     

A cognitive analysis of Cauchy’s conceptions of function,continuity, limit and infinitesimal,with implications for teaching the calculus

In this paper, we use theoretical frameworks from mathematics education and cognitive psychology to analyse Cauchy’s ideas of function, continuity, limit and infinitesimal expressed in his Cours D’Analyse. Our analysis focuses on the development of mathematical thinking from human perception and action into more sophisticated forms of reasoning and proof, offering different insights from those afforded by historical or mathematical analyses. It highlights the conceptual power of Cauchy’s vision and the fundamental change involved in passing from the dynamic variability of the calculus to the modern set-theoretic formulation of mathematical analysis. This offers a re-evaluation of the relationship between the natural geometry and algebra of elementary calculus that continues to be used in applied mathematics, and the formal set theory of mathematical analysis that develops in pure mathematics and evolves into the logical development of non-standard analysis using infinitesimal concepts. It suggests that educational theories developed to evaluate student learning are themselves based on the conceptions of the experts who formulate them. It encourages us to reflect on the principles that we use to analyse the developing mathematical thinking of students, and to make an effort to understand the rationale of differing theoretical viewpoints.     

Institutional and personal meanings of mathematical proof

Although studies on students’ difficulties in producing mathematical proofs have been carried out in different countries, few research workers have focussed their attention on the identification of mathematical proof schemes in university students. This information is potentially useful for secondary school teachers and university lecturers. In this article, we study mathematical proof schemes of students starting their studies at the University of Córdoba (Spain) and we relate these schemes to the meanings of mathematical proof in different institutional contexts: daily life, experimental sciences, professional mathematics, logic and foundations of mathematics. The main conclusion of our research is the difficulty of the deductive mathematical proof for these students. Moreover, we suggest that the different institutional meanings of proof might help to explain this difficulty. This revised version was published online in July 2006 with corrections to the Cover Date.     

小学数学“学习解决生活原型问题”的实践研究

根据新数学课程标准的要求，数学教学必须从学生熟悉的生活情境和感兴趣的事物出发，学生提供观察和操作的机会，使学生在问题中有更多的机会从周围熟悉的事物中学习和理解数学。为了增强学生的应用意识，小学数学应重视生活化的探索，即在教学中从学生的生活经验和已有知识背景出发，用问题解决学习的方式，寻找生活中的数学原型，联系生活实际讲数学，体现数学，用“源于生活、寓于生活、用于生活”的思想，来激发学生学习数学的兴趣，培养学生的逻辑思维能力，让学生真切地体会到数学就在身边。     

Mathematical knowledge and the problem of proof

Every proof is faced with the requirement of proving that the proof is correct, and the proof of the correctness of the proof again meets the same requirement and the proof of the correctness of the correctness of the proof also, etc. In order to escape from an infinite regress into which one is led one has to come down with a purely algorithmic criterion for correctness or to claim that thinking is identical with its subject matter. Whence the preference of number and more generally of conceptualism in pure mathematics. Conceptualism is a kind of nominalism that does not give a realist understanding of mathematics (note that Platonism is not an opponent of nominalism as some seem to believe). The paper presents some examples and reflections intending to hint at the role of formal thought in the process of knowledge growth. It argues that there is no division of labor according to which certain modes of human cognition are associated with certain tasks and certain cognitive roles exclusively. In this connection, the paper claims that the subject matter of mathematical activity is represented within the system of activity by many different means. Mathematics differs in fact from logic in as much as a principle of heterogeneity or of flexible means-objects-relationships is valid. Formalization in contrast brings forward a principle of homogeneity — that like follows like. Every subject matter requires principles homogeneous with itself. The paper tries to draw some conclusions from this difference with respect to the role of formalization within human cognitive development.     

论模糊逻辑的有效性

为了符合经典有效性标准,我国学者王国俊教授和婓道武教授分别构造了模糊形式演绎系统L*和模糊谓词一阶形式系统K*,这反驳了模糊逻辑缺乏逻辑基础的谬论。然而随着现代科学日益进步和逻辑学分支学科的不断发展,经典逻辑过于狭隘的非此即彼的模式及其有效性标准显得越来越过时了。非形式逻辑通过对经典有效性标准的限制和修改,已形成归纳有效性、实质有效性等另类的有效性概念。模糊逻辑用发展了的有效性标准来评估更能体现其理论存在的价值和自身特点。     

Mathematics and Virtual Culture: an Evolutionary Perspective on Technology and Mathematics Education

This paper suggests that from a cognitive-evolutionary perspective, computational media are qualitatively different from many of the technologies that have promised educational change in the past and failed to deliver. Recent theories of human cognitive evolution suggest that human cognition has evolved through four distinct stages: episodic, mimetic, mythic, and theoretical. This progression was driven by three cognitive advances: the ability to ‘represent’ events, the development of symbolic reference, and the creation of external symbolic representations. In this paper, we suggest that we are developing a new cognitive culture: a ‘virtual’ culture dependent on the externalization of symbolic processing. We suggest here that the ability to externalize the manipulation of formal systems changes the very nature of cognitive activity. These changes will have important consequences for mathematics education in coming decades. In particular, we argue that mathematics education in a virtual culture should strive to give students generative fluency to learn varieties of representational systems, provide opportunities to create and modify representational forms, develop skill in making and exploring virtual environments, and emphasize mathematics as a fundamental way of making sense of the world, reserving most exact computation and formal proof for those who will need those specialized skills. This revised version was published online in July 2006 with corrections to the Cover Date.     

让数学思想成为学生的核心数学素养--核心素养视角下的小学数学教学

数学的核心涉及数学思维和数学逻辑两个方面,在现实社会实际应用中,现代计算机技术多依赖于数学二进制算法。学生如果能掌握科学的数学思想,并且将其印入学生的脑海中,学生就会受益,所以教师应当让数学思想成为学生的核心数学素养。     

MATHEMATICS TEACHERS’ CONCEPTIONS OF PROOF: IMPLICATIONS FOR EDUCATIONAL RESEARCH

Current mathematics education reforms devoted to reasoning and proof highlight its importance in pre-kindergarten through grade 12. In order to provide students with opportunities to experience and understand proof, mathematics teachers must have a solid understanding of proofs themselves. In light of this challenge, a growing number of researchers around the world have started to investigate mathematics teachers’ conceptions of proof; however, much more needs to be done. Drawing on lessons learned from research and curricular recommendations from around the world, the main purpose of this paper is to review the literature on elementary and secondary mathematics teachers’ conceptions of proof and discuss international implications.     

浅论反例在高数教学中的作用

该文阐述和强调了反例在高等数学教学中的重要作用:一方面有利于正确理解数学概念、熟练掌握数学定理和数学方法,体会数学的严谨性;另一方面有助于提高学生辩证逻辑思维能力,以期引起师生对教与学反例的重视.     

高等数学哲理性知识教学的调查报告

数学哲理性知识是数学文化宝库中的精品;利用微积分对学生进行数学哲理性知识教育,具有得天独厚的功效.调查结果启示我们:高等数学教学要文理交融,数哲联姻,唯有引导学生让思维的触角延伸到哲学层面,使用矛盾分析法和运用辩证逻辑思维,方可领悟习得数学哲理性知识.揭示数学哲理性知识的思维教学活动的教育价值,在于由这些知识所引发的哲学思考中存在的理性精神教育.     

调动学习数学的“发动机”,促进学生自我学习

数学的逻辑性很强,如果不认真去学习,就会无效学习。所以必须调动学生身上的＂发动机＂,促进学生自我学习。为此,可以采取以下策略教学：创设数学生活情境,使枯燥的数学有趣化;借助现代教育技术,让数学动起来;分层次教学,挖掘学生自我学习的潜力;采取激励措施,长效调动学生的积极性;激发学生主体意识,调动其学习积极性;授之以渔,教给学生自我学习的方法。     

Using structured games to teach early fraction concepts to students who are deaf or hard of hearing

The study focused on the development of the concept of fractions in a group of 11- and 12-year-old students who were deaf or hard of hearing. The approach implemented in the study relied extensively on the use of games with very little formal instruction. It emphasized the development of appropriate language to facilitate an understanding of the notion of fractions through the investigation of concrete materials, pictorial representations, and interactions between students and teacher. The progress achieved by means of this approach is reported, and the implications of developing an understanding of fractions (and mathematics generally) among students who are deaf or hard of hearing are noted.     

小学高年级数学的趣味性教学实践探究

小学数学相对于其他学科来说,学生学习起来具有一定的难度,很多学生在学习的过程中可能会因为对某些数学知识点不理解而对这门课程产生排斥感,而开展小学数学趣味性教学实践活动,不仅可以营造课堂学习的氛围,让学生可以在一个良好的环境下进行学习,而且还可以让学生切实地感受到学习数学知识的乐趣,让学生不再把学习数学知识当作一种负担。为此,教师应该重视趣味性教学,做好课堂教学的工作。     

Making the transition to formal proof

This study examined the cognitive difficulties that university students experience in learning to do formal mathematical proofs. Two preliminary studies and the main study were conducted in undergraduate mathematics courses at the University of Georgia in 1989. The students in these courses were majoring in mathematics or mathematics education. The data were collected primarily through daily nonparticipant observation of class, tutorial sessions with the students, and interviews with the professor and the students. An inductive analysis of the data revealed three major sources of the students' difficulties: (a) concept understanding, (b) mathematical language and notation, and (c) getting started on a proof. Also, the students' perceptions of mathematics and proof influenced their proof writing. Their difficulties with concept understanding are discussed in terms of a concept-understanding scheme involving concept definitions, concept images, and concept usage. The other major sources of difficulty are discussed in relation to this scheme.This article is based on the author's doctoral dissertation completed in 1990 at the University of Georgia under the direction of Jeremy Kilpatrick.     

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.     

动态逻辑的强完全性和不协调性

动态逻辑的证明系统揭示了典范模型的一些比较特殊的特征,即程序的不协调性。在标准的模态逻辑证明系统中,真值引理指在典范模型的世界里真的公式集．正好是由这些公式所组成的极大一致集,该公式集是由哪个世界来识别的,称之为公式协调性,相对于程序来说就称为程序协调性。使用协调的原因在于当公理和程序都是协调的时候,典范模型的语义和证明理论是完全一致的。在此给出了无穷的动态逻辑的证明系统,即命题动态逻辑的一个强完全性的证明系统．是相对比较简单的系统,这使得证明也相对比较简单,但却可以很容易地扩张为其它的模态逻辑。同时证明了该典范模型不是程序协调的．而是公式协调的。