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1.
This paper examines the process through which students learn to make functional use of computer algebra systems (CAS), and
the interaction between that process and the wider mathematical development of students. The result of ‘instrumentalising‘
a device to become a mathematical tool and correspondingly ‘instrumenting’ mathematical activity through use of that tool
is not only to extend students' mathematical technique but to shape their sense of the mathematical entities involved. These
ideas have been developed within a French programme of research – as reported by Artigue in this issue of the journal – which
has explored the integration of CAS – typically in the form of symbolic calculators – into the everyday practice of mathematics
classrooms. The French research –influenced by socio-psychological theorisation of the development of conceptual systems-
seeks to take account of the cultural and cognitive facets of these issues, noting how mathematical norms – or their absence
– shape the mental schemes which students form as they appropriate CAS as tools. Instrumenting graphic and symbolic reasoning
through using CAS influences the range and form of the tasks and techniques experienced by students, and so the resources
available for more explicit codification and theorisation of such reasoning. This illuminates an influential North American
study– conducted by Heid – which French researchers have seen as taking a contrasting view of the part played by technical
activity in developing conceptual understanding. Reconsidered from this perspective, it appears that while teaching approaches
which ‘resequence skills and concepts’ indeed defer – and diminish –attention to routinised skills, the tasks introduced in
their place depend on another –albeit less strongly codified – system of techniques, supporting more extensive and active
theorisation. The French research high lights important challenges which arise in instrumenting classroom mathematical activity
and correspondingly instrumentalising CAS. In particular, it reveals fundamental constraints on human-machine interaction
which may limit the capacity of the present generation of CAS to scaffold the mathematical thinking and learning of students.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
2.
In many mathematical problems, students can feel that the universalityof a conjecture or a formula is validated by their experimentand experience. In contrast, students generally do not feelthat deductive explanations strengthen their conviction thata conjecture or a formula is true. In order to cope up withstudents conviction based only on empirical experienceand to create a need for deductive explanations, we developeda problem-solving activity with technology support intendedto cause cognitive conflict. In this article, we describe theprocess conducted for this activity that led students to contradictionsbetween conjectures and findings. The teacher could create familiarproblem-solving situations and use students naïveinductive approaches to make students think mathematically andestablish the necessity for proof via computer support. 相似文献
3.
Arsalan Wares 《International Journal of Mathematical Education in Science & Technology》2013,44(6):814-819
This note describes two conjectures pertaining to repeated partitioning of an arbitrary triangle. The first conjecture turns out to be true, and hence gives rise to a new, more general, conjecture that is also addressed in this article. Both conjectures can be explored in a dynamic geometry environment. The proofs to the conjectures addressed in this article require knowledge of high school Euclidean geometry. 相似文献
4.
Generalization and proof are defining activities within mathematics, yet the focus of "school" proof has often been on form
over meaning, on established results rather than exploration and discovery. Computer-based microworlds offer opportunities
for students to notice and describe patterns, formulate generalizations, and generate and test mathematics conjectures. This
paper examines the work of a group of middle and high school students who used a microworld for transformation geometry to
investigate the composition of reflections. The students‘conjectures are described in terms of a learning paths chart for
the task, as well as through a detailed analysis of the work of one pair of students. A general scheme for describing informal
exploration and reasoning prior to formal proof is offered, and the role of social support in the learning of proof is discussed.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
5.
Richard M. Grassl Tabitha T. Y. Mingus 《International Journal of Mathematical Education in Science & Technology》2013,44(5):715-723
This article presents teaching ideas designed to support the belief that students at all levels (preservice teachers, majors, secondary and elementary students) need exposure to non-routine problems that illustrate the effective use of technology in their resolution. Such use provides students with rapid and accurate data collection, leading them to sound conjectures, which is a precursor to learning mathematical proof. Students will therefore learn that while technology can be an effective tool for investigating problems, the onus of providing convincing arguments and proofs of their conjectures rests squarely on their shoulders. The paper describes how a diverse group of students took advantage of the power of the TI-92 to enhance their chances of reaching this final stage of proof. A series of mathematical problems are presented and analysed with a keen eye on the appropriate integration of the TI-92. A student survey was used to inform the results. To conclude, several challenging, yet accessible, non-routine problems were completed by students as undergraduate research projects, all using the TI-92 as a laboratory. Although most of the problems presented here have a discrete mathematics flavour, the authors' message is independent of the mathematical topic chosen. 相似文献
6.
We examine issues that arise in students’ making of generalizations about geometrical figures as they are introduced to linear functions. We focus on the concepts of patterns, function, and generalization in mathematics education in examining how 15 third grade students (9 years old) come to produce and represent generalizations during the implementation of two lessons from a longitudinal study of early algebra. Many students scan output values of f(n) as n increases, conceptualizing the function as a recursive sequence. If this instructional route is pursued, educators need to recognize how students’ conceptualizations of functions depart from the closed form expressions ultimately aimed for. Even more fundamentally, it is important to nurture a transition from empirical generalizations, based on conjectures regarding cases at hand, to theoretical generalizations that follow from operations on explicit statements about mathematical relations. 相似文献
7.
The purpose of this study is to explore how students changes through learning to construct mathematical proofs in an inquiry-based middle school geometry class in Korea. Although proof has long been considered as one of the most important aspects of mathematics education, it is well-known that it is one of the most difficult areas of school mathematics for students. The geometry inquiry classroom (GIC) is an experimental class designed to teach geometry, especially focusing on teaching proof, based on students’ own inquiry. Based on a 2-year participant observation in the GIC, this study was conducted to address the following research question: how has students’ practice of mathematical proof been changed through their participation in the GIC? The in-depth analysis of the classroom discourse identified three stages through which the students’ practice of mathematical proof was transformed in the GIC: ‘emergent understanding of proof’, ‘proof learning as a goal-oriented activity’, ‘experiencing proof as the practice of mathematics’. The study found that as learning evolved through these stages, so the mathematics teacher’s role shifted from being an instructor to a mediator of communication. Most importantly, this research showed that the GIC has created a learning environment where students develop their competence in constructing meaningful mathematical proof and grow to be ‘a human who proves’, ultimately ‘a person who playfully engages with mathematics’. 相似文献
8.
Marina Ratner 《Israel Journal of Mathematics》1992,80(1-2):1-31
In this paper I give simple proofs of Raghunathan’s conjectures for SL(2,R). These proofs incorporate in a simplified form some of the ideas and methods I used to prove the Raghunathan’s conjectures
for general connected Lie groups.
Partially supported by the NSF Grant DMS-8701840. 相似文献
9.
Shlomo Vinner 《ZDM》2011,43(2):247-256
The purpose of this paper is to present a view of three central conceptual activities in the learning of mathematics: concept formation, conjecture formation and conjecture verification. These activities also take place in everyday thinking, in which the role of examples is crucial. Contrary to mathematics, in everyday thinking examples are, very often, the only tool by which we can form concepts and conjectures, and verify them. Thus, relying on examples in these activities in everyday thought processes becomes immediate and natural. In mathematics, however, we form concepts by means of definitions and verify conjectures by mathematical proofs. Thus, mathematics imposes on students certain ways of thinking, which are counterintuitive and not spontaneous. In other words, mathematical thinking requires a kind of inhibition from the learners. The question is to what extent this goal can be achieved. It is quite clear that some people can achieve it. It is also quite clear that many people cannot achieve it. The crucial question is what percentage of the population is interested in achieving it or, moreover, what percentage of the population really cares about it. 相似文献
10.
Non-attendance to meaning by students is a prevalent phenomenon in school mathematics. Our goal is to investigate features of instruction that might account for this phenomenon. Drawing on a case study of two high school algebra teachers, we cite episodes from the classroom to illustrate particular teaching actions that de-emphasize meaning. We categorize these actions as pertaining to (a) purpose of new concepts, (b) distinctions in mathematics, (c) mathematical terminology, and (d) mathematical symbols. The specificity of the actions that we identify allows us to suggest several conjectures as to the impact of the teaching practices observed on student learning: that students will develop the belief that mathematics involves executing standard procedures much more than meaning and reasoning, that students will come to see mathematical definitions and results as coincidental or arbitrary, and that students’ treatment of symbols will be largely non-referential. 相似文献
11.
Maria Kordaki Athanasia Balomenou 《International Journal of Computers for Mathematical Learning》2006,11(1):99-135
This study focuses on the constructions in terms of area and perimeter in equivalent triangles developed by students aged 12–15 years-old, using the tools provided by Cabri-Geometry II [Labore (1990). Cabri-Geometry (software), Université de Grenoble]. Twenty-five students participated in a learning experiment where they were asked to construct: (a) pairs of equivalent triangles “in as many ways as possible” and to study their area and their perimeter using any of the tools provided and (b) “any possible sequence of modifications of an original triangle into other equivalent ones”. As regards the concept of area and in contrast to a paper and pencil environment, Cabri provided students with different and potential opportunities in terms of: (a) means of construction, (b) control, (c) variety of representations and (d) linking representations, by exploiting its capability for continuous modifications. By exploiting these opportunities in the context of the given open tasks, students were helped by the tools provided to develop a broader view of the concept of area than the typical view they would construct in a typical paper and pencil environment. 相似文献
12.
Chen Falai 《分析论及其应用》1995,11(2):1-8
This paper proved the following three facts about the Lipschitz continuous property of Bernstein polynomials and Bezier nets
defined on a triangle: suppose f(P) is a real valued function defined on a triangle T, (1) If f(P) satisfies Lipschitz continuous
condition, i.e. f(P)∃LipAα, then the corresponding Bernstein Bezier net fn∃Lip
Asec
αφα, here φ is the half of the largest angle of triangle T; (2) If Bernstein Bezier net fn∃Lip
Bα, then its elevation Bezier net Efn∃Lip
Bα; and (3) If f(P)∃Lip
Aα, then the corresponding Bernstein polynomials Bn(f;P)∃Lip
Asec
αφα, and the constant Asecαφ is best in some sense.
Supported by NSF and SF of National Educational Committee 相似文献
13.
In 1989, Kalai stated three conjectures A, B, C of increasing strength concerning face numbers of centrally symmetric convex polytopes. The weakest conjecture, A, became known as the “3
d
-conjecture.” It is well known that the three conjectures hold in dimensions d≤3. We show that in dimension 4 only conjectures A and B are valid, while conjecture C fails. Furthermore, we show that both conjectures B and C fail in all dimensions d≥5. 相似文献
14.
In different international studies on mathematical achievement East Asian students outperformed the students from Western
countries. A deeper analysis shows that this is not restricted to routine tasks but also affects students’ performance for
complex mathematical problem solving and proof tasks. This fact seems to be surprising since the mathematics instruction in
most of the East Asian countries is described as examination driven and based on memorising rules and facts. In contrast,
the mathematics classroom in western countries aims at a meaningful and individualised learning. In this article we discuss
this “paradox” in detail for Taiwan and Germany as two typical countries from East Asia and Western Europe. 相似文献
15.
Monte J. Zerger 《Mathematical Intelligencer》1998,20(2):5-6
Conclusion I often ponder on my duties as a teacher of the subject I love. I feel I am responsible for more than simply transmitting
knowledge. I wish I could help my students see mathematics from various vantage points. One of these should be from a point
high enough to afford a full, sweeping view of the mathematical valley below—maybe missing the details we strive to convey
in class-but seeing thelandscape of mathematics. Claude Bragdon said, “Mathematics is the handwriting on the human consciousness of the very Spirit of Life itself.” I want
my students to consider that such a bold statement might actually be true. 相似文献
16.
R. V. Gorbachev 《Theoretical and Mathematical Physics》2010,162(1):90-94
We construct a new solution of the superstring equation of motion and show that this solution satisfies two of Sen’s conjectures
and does not require “phantom terms.” 相似文献
17.
In the United States, researchers argue that proof is largely concentrated in the domain of high school geometry, thus providing students a distorted image of what proof entails, which is at odds with the central role that proof plays in mathematics. Despite the centrality of proof, there is a lack of studies addressing how to integrate proof into other mathematical domains. In this article, we discuss a teaching experiment designed to integrate algebra and proof in the high school curriculum. Algebraic proof was envisioned as the vehicle that would provide high school students the opportunity to learn not only about proof in a context other than geometry but also about aspects of algebra. Results from the experiment indicate that students meaningfully learned about aspects of both algebra and proof in that they produced algebraic proofs involving multiple variables and a single parameter, based on conjectures they themselves generated. 相似文献
18.
任韩和李刚在图的最大亏格综述一文"Survey of maximum genus of graphs" [J East China NormUniv Natur Sci, Sep. 2010, No. 5, 1-13] 中,全面地阐述了近30 年来关于图的最大亏格及其相关问题所取得的进展,并提出了如下两个猜想:
猜想1 设G 为简单连通图, 且G 的每条边含在一个三角形K3 中, 则G 是上可嵌入的.
猜想2 设c 为任意的正数, 则存在一个自然数N(c), 使得对每一个图G, 若G 的点数n ≥ N(c), 且最小度δ(G) ≥ cn, 则G 是上可嵌入的.
本文的主要工作是否定上述两个猜想, 同时探讨上述猜想成立的条件且得了一些新结果, 并提出有关进一步研究的问题. 相似文献
19.
Jesse L. M. Wilkins 《School science and mathematics》1999,99(3):117-123
The Cereal Box problem is fascinating for students of ail levels. Rich in mathematical content, this problem offers students the opportunity to collect data, make conjectures, and derive mathematical models. Using Monte Carlo methods, the Cereal Box problem is investigated in this paper, using both an experimental and theoretical framework. This investigation extends previous considerations of the Cereal Box problem. Using empirical data, students can discover patterns and relationships that help them understand the origin of the theoretical solution to the problem. Building on experimental findings, a theoretical model is derived, showing that the expected solution of the Cereal Box problem is formed from the sum of successive geometric series. 相似文献