共查询到20条相似文献，搜索用时 31 毫秒
1.
Shailesh A. Shirali 《Resonance》2013,18(7):630-645
It is sometimes the case in mathematics that the same result can be proved in two or more essentially different ways (a luxury of mathematical riches, you might say). A recurrent theme in this context is the tension between combinatorial proofs and algebraic proofs. Inthis two-part article we describe some results which allow for proofs of both kinds and give us an opportunity to contrast them. We shall see some beautiful examples of both kinds of proofs. In Part I we dwell on results that deal with the binomial coefficients and with Fibonacci numbers. We will not be overly zealous in our usage of the term ‘combinatorial’: we will allow a proof to be so categorized if it is largely combinatorial, with only a ‘small’ component of algebra. However, that small part may well be an essential component of the proof. The simplicity of counting arguments can be deceptive. In recent times, Paul Erd?s has shown just how subtle and powerful such arguments can be, and how far they can reach. 相似文献
2.
In a recent paper (Weber & Mejia-Ramos, Educational Studies in Mathematics, 76, 329–344, 2011), we reported findings from two small-scale interview studies on the reasons why and the ways in which mathematicians read proofs. Based on these findings, we designed an Internet-based survey that we distributed to practicing mathematicians working in top mathematics departments in the USA. Surveyed mathematicians (N?=?118) agreed to a great extent with the interviewed mathematicians in the exploratory studies. First, the mathematicians reported that they commonly read published proofs to gain different types of insight, not to check the correctness of the proofs. Second, they stated that when reading these proofs, they commonly: (a) appeal to the reputation of the author and the journal, (b) study how certain steps in the proof apply to specific examples, and (c) focus on the overarching ideas and methods in the proofs. In this paper, we also report findings from another section of the survey that focused on how participants reviewed proofs submitted for publication. The comparison of participant responses to questions in these two sections of the survey suggests that reading a published proof of a colleague and refereeing a proof for publication are substantially different activities for mathematicians. 相似文献
3.
Juan Pablo Mejia-Ramos Evan Fuller Keith Weber Kathryn Rhoads Aron Samkoff 《Educational Studies in Mathematics》2012,79(1):3-18
Although proof comprehension is fundamental in advanced undergraduate mathematics courses, there has been limited research
on what it means to understand a mathematical proof at this level and how such understanding can be assessed. In this paper,
we address these issues by presenting a multidimensional model for assessing proof comprehension in undergraduate mathematics.
Building on Yang and Lin’s (Educational Studies in Mathematics 67:59–76, 2008) model of reading comprehension of proofs in high school geometry, we contend that in undergraduate mathematics a proof is
not only understood in terms of the meaning, logical status, and logical chaining of its statements but also in terms of the
proof’s high-level ideas, its main components or modules, the methods it employs, and how it relates to specific examples.
We illustrate how each of these types of understanding can be assessed in the context of a proof in number theory. 相似文献
4.
Mohan Chinnappan Madduma B. Ekanayake Christine Brown 《International Journal of Science and Mathematics Education》2012,10(4):865-887
Within the domain of geometry, proof and proof development continues to be a problematic area for students. Battista (2007) suggested that the investigation of knowledge components that students bring to understanding and constructing geometry proofs could provide important insights into the above issue. This issue also features prominently in the deliberations of the 2009 International Commission on Mathematics Instruction Study on the learning and teaching of proofs in mathematics, in general, and geometry, in particular. In the study reported here, we consider knowledge use by a cohort of 166 Sri Lankan students during the construction of geometry proofs. Three knowledge components were hypothesised to influence the students’ attempts at proof development: geometry content knowledge, general problem-solving skills and geometry reasoning skills. Regression analyses supported our conjecture that all 3 knowledge components played important functions in developing proofs. We suggest that whilst students have to acquire a robust body of geometric content knowledge, the activation and the utilisation of this knowledge during the construction of proof need to be guided by general problem-solving and reasoning skills. 相似文献
5.
While proof is central to mathematics, difficulties in the teaching and learning of proof are well-recognised internationally. Within the research literature, a number of theoretical frameworks relating to the teaching of different aspects of proof and proving are evident. In our work, we are focusing on secondary school students learning the structure of deductive proofs and, in this paper, we propose a theoretical framework based on this aspect of proof education. In our framework, we capture students’ understanding of the structure of deductive proofs in terms of three levels of increasing sophistication: Pre-structural, Partial-structural, and Holistic-structural, with the Partial-structural level further divided into two sub-levels: Elemental and Relational. In this paper, we apply the framework to data from our classroom research in which secondary school students (aged 14) tackled a series of lessons that provided an introduction to proof problems involving congruent triangles. Using data from the transcribed lessons, we focus in particular on students who displayed the tendency to accept a proof that contained logical circularity. From the perspective of our framework, we illustrate what we argue are two independent aspects of Relational understanding of the Partial-structural level, those of universal instantiation and hypothetical syllogism, and contend that accepting logical circularity can be an indicator of lack of understanding of syllogism. These findings can inform how teaching approaches might be improved so that students develop a more secure understanding of deductive proofs and proving in geometry. 相似文献
6.
In this paper, we report a study in which nine research mathematicians were interviewed with regard to the goals guiding their
reading of published proofs and the type of reasoning they use to reach these goals. Using the data from this study as well
as data from a separate study (Weber, Journal for Research in Mathematics Education 39:431–459, 2008) and the philosophical literature on mathematical proof, we identify three general strategies that mathematicians employ
when reading proofs: appealing to the authority of other mathematicians who read the proof, line-by-line reading, and modular
reading. We argue that non-deductive reasoning plays an important role in each of these three strategies. 相似文献
7.
The starting point of our reflections is a classroom situation in grade 12 in which it was to be proved intuitively that non-trivial solutions of the differential equationf′=f have no zeros. We give a working definition of the concept of preformal proving, as well as three examples of preformal proofs. Then we furnish several such proofs of the aforesaid fact, and we analyse these proofs in detail. Finally, we draw some conclusions for mathematics in school and in teacher training. 相似文献
8.
Andreas J. Stylianides Gabriel J. Stylianides 《Educational Studies in Mathematics》2009,72(2):237-253
In this article, we focus on a group of 39 prospective elementary (grades K-6) teachers who had rich experiences with proof,
and we examine their ability to construct proofs and evaluate their own constructions. We claim that the combined “construction–evaluation”
activity helps illuminate certain aspects of prospective teachers’ and presumably other individuals’ understanding of proof
that tend to defy scrutiny when individuals are asked to evaluate given arguments. For example, some prospective teachers
in our study provided empirical arguments to mathematical statements, while being aware that their constructions were invalid.
Thus, although these constructions considered alone could have been taken as evidence of an empirical conception of proof,
the additional consideration of prospective teachers’ evaluations of their own constructions overruled this interpretation
and suggested a good understanding of the distinction between proofs and empirical arguments. We offer a possible account
of our findings, and we discuss implications for research and instruction. 相似文献
9.
系统研究数值积分算法基于Java Applet嵌入网页的技术,设计开发Web可视化数值积分软件.该软件实现了对任意初等函数的识别,提供了多种可选择的积分方法,并通过图形展现计算过程.用户通过使用Applet小程序的演示和计算功能,可以更加直观地学习数值积分方法,达到学习软件的交互性、趣味性和直观性的目的.本软件亦可作为数值积分计算器使用,为用户利用数值积分解决实际问题提供方便. 相似文献
10.
AbstractWe review literature relevant to using technology in the teaching/learning of mathematics to highlight four roles of effective technology use: (a) promoting cycles of proof; (b) presenting and connecting multiple representations; (c) supporting case-based reasoning; and (d) serving as a tutee. We then discuss how they intersect with good instruction. Finally, we provide specific examples to illustrate how these roles of technology can be used to maintain the focus of a technology course on mathematics. 相似文献
11.
T. J. Fletcher 《Educational Studies in Mathematics》1976,7(1-2):71-82
Summary The basic properties of nomograms can be proved most easily by metrical methods, but later on the practical problems of nomogram calibration lead naturally to Pappus's theorem. Rearranging the material, we see that Pappus's theorem alone establishes the commutative and associative properties of additive and multiplicative nomograms. Use of Desargues' theorem alone likewise enables proofs to be given of all of these properties, except the commutativity of multiplication. The breakdown of the proof in this case is not sufficient to demonstrate the impossibility of constructing a proof by any other means; nevertheless the material presented indicates the fundamental role of the theorems of Pappus and Desargues, in a context which is much simpler than the usual one, where they are used to set up a coordinate system in the plane. In this way some of the central ideas of Hilbert's Grundlagen can be approached at school level. 相似文献
12.
We propose the spectrum allocation and resource scheduling algorithms in cognitive point to multipoint （PMP） networks with rapid changes of spectrum opportunities and present a media access control （MAC） protocol based on these algorithms. The objective of spectrum allocation is to make efficient use of the spectrum while maintaining the transceiver synchronization on frequency and time in the network. The objective of resource scheduling is to guarantee the quality of service （QoS） requirements of different kinds of connections and to minimize the total energy consumption in the network as well. By sensing only a small set of possible channels in each slot based on the state transition probability of each channel, our spectrum allocation algorithm achieves high spectrum efficiency in the network. The resource scheduling problem is divided into three sub problems and we derive optimal solutions to these problems by greedy algorithm and convex optimization. The simulation results show that our algorithm can make efficient use of the spectrum and the network resources at a cost of low computational complexity. 相似文献
13.
VolGraph-A PC-Based Visualization System for Three-Dimensional Medical Reconstructions 总被引：1，自引：0，他引：1
Thecontinuoustechnicaldevelopmentincomputergraphics ,scientificvisualizationandmedicalscannertechnol ogyhasmadepossiblefastandaccuratethree dimensionalmedicalvisualizationfrommulti slicesoftomographyimagesoverthepastdecade .Traditionaltomographyimagesusedinthree dimensionalreconstructionsaremainlyfromcomputer izedtomography (CT) .Thelatestresearchesextendthedo maintomagneticresonanceimaging (MRI) ,computerizedto mographyangiography (CTA) ,singlephotoemissiontomogra phy(SPET) ,positronemiss… 相似文献
14.
A new method in highway route design: joining circular arcs by a single C-Bézier curve with shape parameter 总被引：1，自引：1，他引：0
We constructed a single C-Bezier curve with a shape parameter for G^2 joining two circular arcs. It was shown that an S-shaped transition curve, which is able to manage a broader scope about two circle radii than the Bezier curves, has no curvature extrema, while a C-shaped transition curve has a single curvature extremum. Regarding the two kinds of curves, specific algorithms were presented in detail, strict mathematical proofs were given, and the effectiveness of the method was shown by examples This method has the following three advantages： （1） the pattern is unified; （2） the parameter able to adjust the shape of the transition curve is available; （3） the transition curve is only a single segment, and the algorithm can be formulated as a low order equation to be solved for its positive root. These advantages make the method simple and easy to implement. 相似文献
15.
AbstractWe present a mathematical activity called graph talks as a new pedagogical routine to intertwine social justice issues and mathematics. Adapted from number talks, graph talks involve students analyzing, interpreting, and discussing real-life data represented in graphs. Graphs may be strategically selected to both highlight a relevant social justice issue while also reinforcing the mathematics content of the course. We report on experiences using graph talks in undergraduate mathematics content courses for future teachers in the USA, and provide examples of the undergraduates doing mathematics while analyzing the social justice context of the graphs. 相似文献
16.
通过对图、完全图和正则图概念的介绍,详细地描述了图嵌入的方法,同时对主成分分析、线性鉴别分析、局部保持投影、保持近邻嵌入、L1图及其嵌入等经典的特征提取算法进行了详细的代数推导,列出了详细的推导过程,得出这些经典算法可以用图嵌入理论来解释的结论,最后得出特征提取算法的核心在于算法的图构造. 相似文献
17.
本文详细介绍了基于寄存器分配的三种软件水印算法,QP,QPS,QPI。这三种算法都是通过为冲突图添加边的方式在程序中嵌入水印的。根据图染色的寄存器分配理论,我们提出了一种新的软件水印算法一二次染色算法（STC）,此算法并不需要添加任何额外的边,只是通过为图中的部分顶点二次着色方式来嵌入水即的。与前面三种算法比较,STC算法更简洁,更有效。 相似文献
18.
19.
This paper presents a novel approach to consider optimal multi-degree reduction of Bézier curve with G1-continuity. By minimizing the distances between corresponding control points of the two curves through degree raising, optimal approximation is achieved. In contrast to traditional methods, which typically consider the components of the curve separately, we use geometric information on the curve to generate the degree reduction. So positions and tangents are preserved at the two endpoints. For satisfying the solvability condition, we propose another improved algorithm based on regularization terms. Finally, numerical examples demonstrate the effectiveness of our algorithms. 相似文献
20.
合理有效地管理实验设备有利于提高设备的利用率,现将时间图查询用于实验设备的管理,可以丰富查询的语义,提高设备的查询效率。将设备的使用情况抽象成一个大的时间图,将用户的查询请求转换为一个查询图,利用图匹配技术查询出相关的结果。为实现查询图的匹配,提出了3种相关算法:朴素匹配算法(NM)、基于BFS的点匹配算法(BVM)和拓扑剪枝匹配算法(TPM)。在TPM算法中设计了2种索引:TV-索引和TE-索引,分别用于快速定位节点和边上的关系,并从结构和语义两个角度对匹配过程进行了剪枝。最后,设计了对比实验,通过实验验证了3种算法的性能。 相似文献