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1.
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.  相似文献   

2.
The aim of this study is to investigate students’ conceptions about proof in mathematics and mathematics teaching. A five‐point Likert‐type questionnaire was administered in order to gather data. The sample of the study included 33 first‐year secondary school mathematics students (at the same time student teachers). The data collected were analysed and interpreted using the methods of qualitative and quantitative analysis. The results have revealed that the students think that mathematical proof has an important place in mathematics and mathematics education. The students’ studying methods for exams based on imitative reasoning which can be described as a type of reasoning built on copying proof, for example, by looking at a textbook or course notes proof or through remembering a proof algorithm. Moreover, they addressed to the differences between mathematics taught in high school and university as the main cause of their difficulties in proof and proving.  相似文献   

3.
Proof and reasoning are fundamental aspects of mathematics. Yet, how to help students develop the skills they need to engage in this type of higher-order thinking remains elusive. In order to contribute to the dialogue on this subject, we share results from a classroom-based interpretive study of teaching and learning proof in geometry. The goal of this research was to identify factors that may be related to the development of proof understanding. In this paper, we identify and interpret students' actions, teacher's actions, and social aspects that are evident in a classroom in which students discuss mathematical conjectures, justification processes and student-generated proofs. We conclude that pedagogical choices made by the teacher, as manifested in the teacher's actions, are key to the type of classroom environment that is established and, hence, to students' opportunities to hone their proof and reasoning skills. More specifically, the teacher's choice to pose open-ended tasks (tasks which are not limited to one specific solution or solution strategy), engage in dialogue that places responsibility for reasoning on the students, analyze student arguments, and coach students as they reason, creates an environment in which participating students make conjectures, provide justifications, and build chains of reasoning. In this environment, students who actively participate in the classroom discourse are supported as they engage in proof development activities. By examining connections between teacher and student actions within a social context, we offer a first step in linking teachers' practice to students' understanding of proof.  相似文献   

4.
This article reports an investigation into how students of a mathematics course for prospective secondary mathematics teachers in England talk about the notion of ‘understanding mathematics in depth’, which was an explicit goal of the course. We interviewed eighteen students of the course. Through our social practice frame and in the light of a review of the literature on mathematical knowledge for teaching, we describe three themes that weave through the students’ talk: reasoning, connectedness and being mathematical. We argue that these themes illuminate privileged messages in the course, as well as the boundary and relationship between mathematical and pedagogic content knowledge in secondary mathematics teacher education practice.  相似文献   

5.
Gila Hanna 《Interchange》2000,31(1):21-33
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.  相似文献   

6.
数学课堂规范包括支配课堂的一般社会规范和围绕特定数学交互的数学规范.数学课堂规范的特征有:数学课堂规范是"预成性"和"生成性"统一的结果;学生数学观念影响数学课堂规范的发展;社会规范和数学课堂规范之间有差异;创生数学课堂规范就是创造一种信任和尊重的学习气氛.做数学的新规范和标准也需面对传统教学方式的桎梏,学生数学观的局限性,以及数学参与公正性等的挑战性和复杂性。  相似文献   

7.
要教好线性代数课程,其核心问题就是要通过课堂教学,使学生理解相关的数学知识;训练和培养学生的思维能力以及数学交流能力;帮助学生寻找新旧知识之间的内在联系,使知识系统化;在巩固已有知识的基础上,让学生自己去发现新知识.要实现这一教学目标,训练学生掌握"数学证明"的概念和在实践中的应用至关重要.传统的做法往往是通过"定义一引理一证明一定理一证明一推论"这种复杂的、程序化方法来进行训练的.由于在初等代数课程中,学生很少接受严格数学证明的训练,所以这种俗套的做法成效甚微.相反,如果把线性代数的主题和概念用一种完全合理的探究式方法来引入,那么数学证明的概念和架构将牢固植根于学生的头脑,并且这种思维习惯将对他们后续课程的学习和掌握公理化推理方法都会有很大帮助.  相似文献   

8.
In this paper we describe an ontological and semiotic model for mathematical knowledge, using elementary combinatorics as an example. We then apply this model to analyze the solving process of some combinatorial problems by students with high mathematical training, and show its utility in providing a semiotic explanation for the difficulty of combinatorial reasoning. We finally analyze the implications of the theoretical model and type of analysis presented for mathematics education research and practice.  相似文献   

9.
This study investigated differences between the US and Finland in terms of how students’ attitude is related to mathematical reasoning skills through the Trends in International Mathematics and Science Study (TIMSS) 2011. Attitude towards mathematics was observed via 3 TIMSS contextual variables: liking mathematics, valuing mathematics, and confidence in mathematics. Scores for mathematical reasoning were collected from the TIMSS 2011 database. We used hierarchical linear modelling to construct multilevel models with interactions of the attitude variables. Findings showed that confidence in mathematics had the strongest positive relationships with mathematical reasoning in both countries. Finnish students generally reported stronger positive relationships between confidence in mathematics and reasoning than US students. Strong relationships between confidence and reasoning remained visible when examining valuing and liking mathematics. Findings provide important implications regarding the complex interactions between attitude towards mathematics and reasoning, critical for mathematics educators and policymakers to consider in an increasingly competitive international environment.  相似文献   

10.
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.  相似文献   

11.
In this paper, we present an analytic framework for investigating expert mathematical learning as the process of building a network of mathematical resources by establishing relationships between different components and properties of mathematical ideas. We then use this framework to analyze the reasoning of ten mathematicians and mathematics graduate students that were asked to read and make sense of an unfamiliar, but accessible, mathematical proof in the domain of geometric topology. We find that experts are more likely to refer to definitions when questioning or explaining some aspect of the focal mathematical idea and more likely to refer to specific examples or instantiations when making sense of an unknown aspect of that idea. However, in general, they employ a variety of types of mathematical resources simultaneously. Often, these combinations are used to deconstruct the mathematical idea in order to isolate, identify, and explore its subcomponents. Some common patterns in the ways experts combined these resources are presented, and we consider implications for education.  相似文献   

12.
Mathematics teachers play a unique role as experts who provide opportunities for students to engage in the practices of the mathematics community. Proof is a tool essential to the practice of mathematics, and therefore, if teachers are to provide adequate opportunities for students to engage with this tool, they must be able to validate student arguments and provide feedback to students based on those validations. Prior research has demonstrated several weaknesses teachers have with respect to proof validation, but little research has investigated instructional sequences aimed to improve this skill. In this article, we present the results from the implementation of such an instructional sequence. A sample of 34 prospective secondary mathematics teachers (PSMTs) validated twelve mathematical arguments written by high school students. They provided a numeric score as well as a short paragraph of written feedback, indicating the strengths and weaknesses of each argument. The results provide insight into the errors to which PSMTs attend when validating mathematical arguments. In particular, PSMTs’ written feedback indicated that they were aware of the limitations of inductive argumentation. However, PSMTs had a superficial understanding of the “proof by contradiction” mode of argumentation, and their attendance to particular errors seemed to be mediated by the mode of argument representation (e.g., symbolic, verbal). We discuss implications of these findings for mathematics teacher education.  相似文献   

13.
Changing perspectives on mathematics teaching and learning resulted in a new generation of mathematics textbooks, stressing among others the importance of mathematical reasoning and problem-solving skills and their application to real-life situations. The article reports a study that investigates to what extent the reform-based ideas underlying these mathematical textbooks impact the current teaching of mathematics. Two problem-solving lessons were videotaped in 10 sixth-grade classrooms and a coding scheme was developed to analyze these lessons with regard to three aspects of the classroom culture that are assumed to enhance students’ mathematical beliefs and problem-solving competencies: (1) the classroom norms that are established, (2) the instructional techniques and classroom organization forms, and (3) the set of tasks students are confronted with. Two instruments were administered to measure students’ beliefs about learning mathematical word problem solving, and to assess their problem-solving processes and skills. The results indicate that some reform-based aspects seemed to be easier to implement (e.g., a strong focus on heuristic skills, embedding tasks in a realistic context) than others (e.g., the use of group work, an explicit negotiation of appropriate social norms).  相似文献   

14.
Teachers of mathematics orchestrate opportunities for interactions between learners and subject matter. Therefore, mathematics teachers need rich, multidimensional content knowledge for teaching mathematics, which incorporates knowledge of the subject matter, students, and teaching. Studying this mathematical knowledge for teaching (MKT) necessitates more than a unidirectional assessment. In this study, the mathematical knowledge for teaching reasoning and proving of two secondary mathematics teachers was investigated through classroom observations and clinical assessments. Results indicate that using MKT as a frame for examining classroom practice, in addition to assessing the MKT a teacher possesses in a clinical setting, provides an in-depth and innovative method for investigating MKT. The comparison of the two cases also identifies student positioning as a key mediating factor between MKT and opportunities to learn. Implications for using MKT as a lens for examining practice in teacher education are discussed.  相似文献   

15.
《初中数学课程标准(2011版)》指出,数学课程能使学生掌握必备的基础知识和基本技能,培养学生的抽象思维和推理能力,培养学生的创新意识和实践能力数学的发散性思维能力是"问题解决"的基础,是培养数学推理能力和创新意识前提要求。数学发散性思维作为用学科自身的品质陶冶人、启迪人、充实人。"问题解决"是人的高级数学思维。高级思维的学习,可以使学生充分享受思维的快乐,可以让思维自由飞翔。本文就初中数学发散思维的培养谈几点体会,通过创设问题情景、设置开放性试题、发挥学科优势等教学策略,着力培养初中学生的数学发散性思维能力,实现有效教学。  相似文献   

16.
Separate tests of mathematics skills, proportions and translations between words, and mathematical expression given the first week of class were correlated with performance for students who completed a college physics course (completes) and students who dropped the course (drops). None of the measures used discriminated between completes and drops as groups. However, the correlations between score on the test of math skills and on both of the measures involving mathematical reasoning (proportions, and translations) were dramatically different for the two groups. For the completes, these correlations were slightly negative, but not significant. For the drops, the correlation was positive and signficant at the p < 0.01 level. This suggests the possibility that the students who complete the course tend to have independent cognitive skills for the “mechanical” mathematical operations and for questions requiring some degree of reasoning, while, in contrast, the same skills for students at high risk for dropping overlap significantly. The study also found that when students are given the results of mathematics skills tests in a diagnostic mode, with feedback on specific areas of weakness and time to remediate with self study, the correlation between mathematics and physics is lower than previously reported values.  相似文献   

17.
Particularly in mathematics, the transition from school to university often appears to be a substantial hurdle in the individual learning biography. Differences between the characters of school mathematics and scientific university mathematics as well as different demands related to the learning cultures in both institutions are discussed as possible reasons for this phenomenon. If these assumptions hold, the transition from school to university could not be considered as a continuous mathematical learning path because it would require a realignment of students’ learning strategies. In particular, students could no longer rely on the effective use of school-related individual resources like knowledge, interest, or self-concept. Accordingly, students would face strong challenges in mathematical learning processes at the beginning of their mathematics study at university. In this contribution, we examine these assumptions by investigating the role of individual mathematical learning prerequisites of 182 first-semester university students majoring in mathematics. In line with the assumptions, our results indicate only a marginal influence of school-related mathematical resources on the study success of the first semester. In contrast, specific precursory knowledge related to scientific mathematics and students’ abilities to develop adequate learning strategies turn out as main factors for a successful transition phase. Implications for the educational practice will be discussed.  相似文献   

18.
We were interested in exploring the extent to which advanced mathematics lecturers provide students with opportunities to play a role in considering or generating course content. To do this, we examined the questioning practices of 11 lecturers who taught advanced mathematics courses at the university level. Because we are unaware of other studies examining advanced mathematics lecturers’ questioning, we first analyzed the data using an open coding scheme to categorize the types of content lecturers solicited and the opportunities they provided students to participate in generating course content. In a second round of analysis, we examined the extent to which lecturers provide students with opportunities to generate mathematical contributions and to engage in reasoning that researchers have identified as important in advanced mathematics. Our findings highlight that, although lecturers asked many questions, lecturers did not provide substantial opportunities for students to participate in generating mathematical content and reasoning. Additionally, we provide several examples of lecturers providing students with some opportunities to generate important contributions. We conclude by providing implications and areas for future research.  相似文献   

19.
This study examines adult students enrolled in pipe trades preapprenticeship training to identify distinguishing features of the mathematical activity within their program and sources of their mathematics-related difficulties. Two closely related sociocultural perspectives, specifically cultural historical activity theory and the theory of knowledge objectification, frame this investigation. Like workplace mathematics practices reported elsewhere, mathematics within this preapprenticeship program was inextricably tied to the efficient production of objects of workplace activity, in this case, the design and fabrication of a limited number of well-defined objects of pipe trades production. This form of mathematical activity was also mediated intensely by semiotic tools and norms of practice specific to the pipe trades. The contribution of this study is the finding that the great majority of the mathematics difficulties encountered by students can be attributed to their novice levels of awareness of the objects of pipe trades production and related ways of working rather than the mathematics understandings that they brought to this endeavor. Implications are identified for the teaching of mathematics in skilled trades training generally, developing students’ mathematical subjectivities within trades training, and the mathematics preparation of secondary school students who will be entering skilled trades. Questions are also raised for further research.  相似文献   

20.
在当前数学实践中,数学知识(如果有这样的知识的话)是通过在定义和公理的基础上证明定理来获得的.问题在于该怎样理解证明中所得到的东西是如何构成知识的,具体而言,即是要给出一个关于数学真理和数学知识的统一的解释,该解释能够揭示两者的内在联系.此处的困难是,根据贝纳塞拉夫的为人熟知的论证,由于塔斯基语义学认为真与对象的联系(通过单称词项或通过量词)是不可消去的,因此在数学中无法将塔斯基语义学与完整的认识论相结合:数学知识要么是通过证明得到的,这种情况下数学知识与数学对象是无关的,因此我们就无法解释数学真理;要么数学对象是数学真理的构件,从而数学知识不是通过证明得到的,这种情况下我们就无从理解数学知识.接着,本文通过一系列阶段,将这些困难一直追溯到最基本的逻辑观念,即将之看作形式的和纯粹解释性的:如果数学是从概念出发仅仅使用逻辑的推理实践,依照康德,那么数学应该是分析的,也即,仅仅是解释性的,根本就不是通常意义上的知识.我认为,这对数学真理是真正困难的问题.本文概括了四种回应,其中仅有一个有希望解决我们的困难,也即皮尔斯和弗雷格的回应.根据他们的方案,逻辑是科学,因此是实验性的和可错的;符号语言是有内容的,尽管并不涉及与任何对象的关联;证明是构成性的,因此是富于产出的过程.通过充分发展这些观点,我们将有可能最终解决数学真理的问题.  相似文献   

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