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1.
In this theoretical paper, we present a framework for conceptualizing proof in terms of mathematical values, as well as the norms that uphold those values. In particular, proofs adhere to the values of establishing a priori truth, employing decontextualized reasoning, increasing mathematical understanding, and maintaining consistent standards for acceptable reasoning across domains. We further argue that students’ acceptance of these values may be integral to their apprenticeship into proving practice; students who do not perceive or accept these values will likely have difficulty adhering to the norms that uphold them and hence will find proof confusing and problematic. We discuss the implications of mathematical values and norms with respect to proof for investigating mathematical practice, conducting research in mathematics education, and teaching proof in mathematics classrooms.  相似文献   

2.
Making the transition to formal proof   总被引：1，自引：0，他引：1  
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.  相似文献   

3.
Many countries are revising their mathematics curriculum in order to elevate the role of proof and argumentation at all school levels and for all student groups. Yet, we have very little research on how proof-related competences are aimed to be developed in the mathematics curricula of different countries in Grades 1 to 12. This article contributes to filling this gap by analysing and comparing three countries’ curricula from the perspective of developmental proof. For this purpose, we created an analytical frame of proof-related competences that could be connected to the development of students’ understanding and skills concerning argumentation and mathematical proof. The analysis reveals three quite different trajectories with specific characteristics, shortcomings and strengths.  相似文献   

4.
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   

5.
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   

6.
What patterns can be observed among the mathematical arguments above-average students find convincing and the strategies these students use to learn new mathematical concepts? To investigate this question, we gave task-based interviews to eleven female students who had performed well in their college-level mathematics courses, but who differed in the number of proof-oriented courses each had taken. One interview was designed to elicit expressions of what students find convincing. These expressions were categorized according to the proof schemes defined by Harel and Sowder (1998). A second interview was designed to elicit expressions of what strategies students use to learn a mathematical concept from its definition, and these expressions were classified according to the learning strategies described by Dahlberg and Housman (1997). A qualitative analysis of the data uncovered the existence of a variety of phenomena, including the following: All of the students successfully generated examples when asked to do so, but they differed in whether they generated examples without prompting and whether they successfully generated examples when it was necessary to disprove conjectures. All but one of the students exhibited two or more proof schemes, with one student exhibiting four different proof schemes. The students who were most convinced by external factors were unsuccessful in generating examples, using examples, and reformulating concepts. The only student who found an examples-based argument convincing generated examples far more than the other students. The students who wrote and were convinced by deductive arguments were successful in reformulating concepts and using examples, and they were the same set of students who did not generate examples spontaneously but did successfully generate examples when asked to do so or when it was necessary to disprove a conjecture.  相似文献   

7.
As a key objective, secondary school mathematics teachers seek to improve the proof skills of students. In this paper we present an analytic framework to describe and analyze students' answers to proof problems. We employ this framework to investigate ways in which dynamic geometry software can be used to improve students' understanding of the nature of mathematical proof and to improve their proof skills. We present the results of two case studies where secondary school students worked with Cabri-Géeomèetre to solve geometry problems structured in a teaching unit. The teaching unit had theaims of: i) Teaching geometric concepts and properties, and ii) helping students to improve their conception of the nature of mathematical proof and to improve their proof skills. By applying the framework defined here, we analyze students' answers to proof problems, observe the types of justifications produced, and verify the usefulness of learning in dynamicgeometry computer environments to improve students' proof skills.This revised version was published online in September 2005 with corrections to the Cover Date.  相似文献   

8.
As a key objective, secondary school mathematics teachers seek to improve the proof skills of students. In this paper we present an analytic framework to describe and analyze students' answers to proof problems. We employ this framework to investigate ways in which dynamic geometry software can be used to improve students' understanding of the nature of mathematical proof and to improve their proof skills. We present the results of two case studies where secondary school students worked with Cabri-Géeomèetre to solve geometry problems structured in a teaching unit. The teaching unit had theaims of: i) Teaching geometric concepts and properties, and ii) helping students to improve their conception of the nature of mathematical proof and to improve their proof skills. By applying the framework defined here, we analyze students' answers to proof problems, observe the types of justifications produced, and verify the usefulness of learning in dynamicgeometry computer environments to improve students' proof skills.  相似文献   

9.
While many semiotic and cognitive studies on learning mathematics have focused primarily on students, this study focuses mainly on teachers, by seeking to bring to light their awareness of the semiotic and cognitive aspects of learning mathematics. The aim is to highlight the degree of awareness that teachers show about: (1) the distinction between what the institution (school, university, society, etc.) proposes as a mathematical object (not in itself but as the content to be learned) and one of its semiotic representations; (2) the different aspects of a semiotic representation that the student able to handle the representation and the student who handles the representation with difficulty may focus on; (3) the semiotic conflicts generated by the contents of semiotic representations that are similar to each other in some respect. For this purpose, in this study, the semio-cognitive approach introduced by Raymond Duval was complemented with the semiotic-interpretative approach of the Peircean tradition. By embracing the pragmatist research paradigm, the methodology was based on the research questions, which guided the selection of the research methods within a qualitatively driven mixed methods design. The research results clearly show the need for a review of professional teacher training programs, as regards the role the semiotic handling plays in the cognitive construction of the mathematical objects and the learning assessment.  相似文献   

10.
College readiness of students and the effectiveness of remedial mathematics courses have been under consideration for the last two decades. There is a considerable misalignment between the expectations of students regarding secondary education and those regarding higher education. Information about current expectations and perspectives of college mathematics faculty who have to deal with this gap is missing in the literature. This study explores college readiness of first-year students and topics that they need to have mastered before entering college. A survey was disseminated to college/university mathematics faculty throughout the US (48 states) whose email addresses were shown on their institutional webpages, and data were gathered from 737 faculty. The survey instrument includes scaled items reflecting the Common Core State Standards and free response items. The scaled items are divided into six subscales: Basics, Algebra, Functions, Geometry, Statistics and Probability, and Reasoning and Generalisation. Faculty responses are categorised and statistically analysed with respect to types of institution, position titles of the participants and types of course offered by those institutions. Findings indicate that faculty view first-year students as having poor mathematical ability in terms of what they consider to be important topics for college preparation. Faculty also agree that students need remediation, which, in its current state, is not sufficient. Implications of these results for further research and practice are discussed.  相似文献   

11.
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   

12.
This study aims to investigate a construct of reading comprehension of geometry proof (RCGP). The research aims to investigate (a) the facets composing RCGP, and (b) the structure of these facets. Firstly, we conceptualize this construct with relevant literature and on the basis of the discrimination between the logical and the epistemic meanings of an argument, then assemble the content of RCGP from literature and propose a hypothetical model of RCGP. Secondly, mathematicians and mathematics teachers are interviewed for their ideas on reading mathematical proof in order to enrich the content of RCGP. Adapting the phases of reading comprehension in language, the content of RCGP is classified into six facets. Lastly, these facets are structured using the hypothetical model and then justified by students’ performance in the facets of RCGP using the multidimensional scaling method. The results sustain that the structure of facets can be characterized by this conceptualized model.  相似文献   

13.
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   

14.
In this paper, we explore the relationship between learners' actions, visualisations and the means by which these are articulated. We describe a microworld, Mathsticks, designed to help students construct mathematical meanings by forging links between the rhythms of their actions and the visual and corresponding symbolic representations they developed. Through a case study of two students interacting with Mathsticks, we illustrate a view of mathematics learning which places at its core the medium of expression, and the building of connections between different mathematisations rather than ascending to hierarchies of decontextualisation.This revised version was published online in September 2005 with corrections to the Cover Date.  相似文献   

15.
In this paper, we aim to contribute to the discussion of the role of the human body and of the concrete artefacts and signs created by humankind in the constitution of meanings for mathematical practices. We argue that cognition is both embodied and situated in the activities through which it occurs and that mathematics learning involves the appropriation of practices associated with the sets of artefacts that have historically come to represent the body of knowledge we call mathematics. This process of appropriation involves a coordination of a variety of the semiotic resources??spoken and written languages, mathematical representation systems, drawings, gestures and the like??through which mathematical objects and relationships might be experienced and expressed. To highlight the connections between perceptual activities and cultural concepts in the meanings associated with this process, we concentrate on learners who do not have access to the visual field. More specifically, we present three examples of gesture use in the practices of blind mathematics students??all involving the exploration of geometrical objects and relationships. On the basis of our analysis of these examples, we argue that gestures are illustrative of imagined reenactions of previously experienced activities and that they emerge in instructional situations as embodied abstractions, serving a central role in the sense-making practices associated with the appropriation of mathematical meanings.  相似文献   

16.
The aim of the research presented in this paper is to contribute to our knowledge about problem solving in mathematics. My purpose in this paper is to compare, from this point of view, two very different institutions in the French tertiary education system, with the intention to interpret the chronic inequality of performance in problem solving between populations of mathematics students coming from these institutions. Problem solving knowledge and skills are not an explicit objective of teaching and their development depends largely on the student's private mathematical activity. This hypothesis is the reason why the inquiry aims at comparing mathematics students' ways of working as they study in both institutions. The results of the research are interpreted, on the institutional level, as effects of differences between the two teaching systems. This revised version was published online in July 2006 with corrections to the Cover Date.  相似文献   

17.
Abstract

School–university partnerships also known as professional development school (PDS) partnerships provide potential for universities and schools to establish partnerships that can benefit university faculty, school teachers, university students, and school students. This study examines the impact of a PDS partnership in which the author served as a school-based mathematics coach for two years in a high-need elementary school. Data sources included interviews, surveys, and field notes from classroom observations. Inductive qualitative analyses which were situated in a multi-level framework for researching professional development found that teachers posed more cognitively demanding mathematical tasks and high-level questions in year two compared to year one of this project. Further, student achievement was noted on both state-wide and district-created assessments. Also teachers reported that the school-based approach to professional development led to some teachers taking on more informal leadership roles to support their colleagues’ mathematics instruction. Implications for school-based learning opportunities across the world include the need to establish specific university–school partnerships, and carefully designing research studies to examine the impact of these learning opportunities.  相似文献   

18.
ABSTRACT

In higher education, engineering students have to be prepared for their future jobs, with knowledge but also with several soft skills, among them creativity. In this paper, we present a study carried on with 128 engineering undergraduate students on their understanding of mathematical creativity. The students were in the first year of different engineering first degrees in a north-eastern Portuguese university and we analysed the content of their texts for the question ‘What do you understand by mathematical creativity?’. Data collection was done in the first semester of the academic years 2014/2015 and 2016/2017 in a Linear Algebra course. The results showed that ‘problem solving’ category had the majority of the references in 2014/2015, but not in the academic year 2016/2017 were ‘involving mathematics’ category had the majority. This exploratory study pointed out for ‘problem solving’ and ‘involving mathematics’ categories and gave us hints for teaching mathematics courses in engineering degrees.  相似文献   

19.
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.  相似文献   

20.
In this article, we present the results of the administration of a questionnaire designed to evaluate the understanding that civil engineering students have of the antiderivative. The questionnaire was simultaneously administered to samples of Mexican and Colombian students. For the analysis of the answers, we used some theoretical and methodological notions provided by the theoretical model known as Onto-Semiotic Approach (OSA) to mathematical cognition and instruction. The results revealed the meanings of the antiderivative that are more predominantly used by civil engineering students. Also, the comparison between the mathematical activity of Mexican and Colombian students provides information that allows concluding that the meanings mobilized could be shared among their communities and are not particular of their classroom or university.  相似文献   

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