化无效作业为有效教学的经历与感悟

1问题背景高三数学复习阶段“作业布置与反馈”环节至关重要,能否有效促进学生理解和应用所学知识和方法是评判作业是否有效的重要依据．同时摒弃题海战术实现减负增效又是势在必行的做法．教师能否保证每一次的作业都绝对有效;面对不同层次的学生,教师能否及时感知作业的正确性;一旦发现低效甚至无效的作业,能否采取适当的补救措施;能否抓住和利用低效或失效作业的生成元素,深入挖掘出利于学生学习和发展的资源;这些问题都是每位高三教师十分关注的．     

圆的重要定理在椭圆和双曲线上的推广

读文[1]，有两点想法：一是圆中的垂径定理能否推广；二是这些定理能否推广到双曲线．下面结合自己思考的一些结果，对文[1]的研究成果作一点补充．     

古典概率模型中样本空间的选择

古典概型是高中数学概率学习的核心．在古典概率的计算过程中，样本空间的选择是关键一环，主要表现在“能否正确选择样本空间”和“能否选择较小样本空间”两方面．在理解题意的过程中是否注重这两方面的思考，将决定解题的成功与高效．本文举例说明，供读者参考。     

几何概率模型求解的几种典型方法

几何概型是新教材必修3《概率》一章中新增加的内容．几何概率模型即每个事件发生的概率只与构成该事件区域的长度（面积或体积）成比例．对于一个具体的问题能否用几何概率模型公式计算其概率,关键是能否将问题几何化,从建立的几何模型入手,来解决概率问题．此类问题由于综合性强、灵活性大,解题时感到无从下手．本文列举几例谈解决此类问题的典型方法．     

关于变式题

关于变式题武汉市武昌区中学教研室林友智武汉市洪山区中学教研室罗焕忠一、问题的提出变式，指的是相对于某种规范（标准）形式的变化形式．随着教学研究的深入，人们普遍认识到，能否正确运用变式，实在是能否有效地提高教育质量的一个关键问题．先看两个例子．例１（范...     

选准提问时机，提升启发效率——谈高中数学课堂提问时机的把握

<正>课堂提问一直都是教师启发学生的重要手段，然而启发效率如何，并不在于提问的次数，而在于问题能否有效激起学生的探索热情，能否驱动学生的思维向着更深层次发展，能否引领学生按照正确的思路来分析问题．因此，为了让问题更好地启发学生，在高中数学课堂上教师要把握好提问时机，结合学生的具体情况合理提出问题．     

站在数学思想的层面看问题解决——以不等式恒成立问题为例

数学思想是解题的航标,问题的解决能否清晰酣畅得心应手,主要是看对解题的思想方法能否融会贯通,用之于润物细无声的境界．本文仅就08、09年高考中出现的部分不等式恒成立的问题,谈一谈如何站在数学思想的层面看待这些问题,以期在解题中更好地领会重要的数学思想,增强认识问题的理性．     

学会揭示问题本质

同学们知道，要解决一个数学问题，在于能否把这个数学问题“看破”．而所谓的“看破”，就是把握解决问题的核心与关键，也即揭示问题的本质．     

ψ-序压缩算子的不动点定理

2005年,张宪在Banach空间中通过其中的锥所定义的半序引进了序压缩算子,证明了几个相应的定理．但是在一般的度量空间中,能否定义序压缩算子,能否得到类似的结论呢？本文在度量空间X中,通过X上的泛函ψ-所定义的半序,引进了ψ--序压缩算子,并且得到了相应的不动点定理．     

一道高考题的探究性学习

探究性学习能否富有成效地开展,一个很重要的原因是选取适宜的探究问题．正如前苏联数学家奥加涅相所说：“很多习题潜在着进一步扩展其数学功能和教育功能的可行性．”     

Mathematics instruction and the tablet PC

The use of tablet PCs in teaching is a relatively new phenomenon. A cross between a notebook computer and a personal digital assistant (PDA), the tablet PC has all of the features of a notebook with the additional capability that the screen can also be used for input. Tablet PCs are usually equipped with a stylus that allows the user to write on the screen. Handwriting recognition software converts this input into text for use with software such as internet browsers and email programs. As an educational tool, two of the most important features of the tablet PC are annotation and wireless communication. The annotation feature allows the user to write on almost any document much as one would annotate a printout of the same document. The wireless communication feature allows tablet PCs to share information with one another. The advantages of these features and their impact on the Murray State University (MSU) classroom will be discussed in the evaluation section.     

卡托普利缓释片释放过程的灰色数学模型

目的:用灰色理论研究卡托普利缓释片的体外释放过程.方法:采用羧甲基纤维素钠为骨架材料制备缓释片,通过体外释放试验,根据灰色数学模型,预测卡托普利缓释片的体外释放过程.结果:预测值与实测值的平均绝对误差E为0.532,平均相对误差为1.059%.结论:为卡托普利缓释片的临床合理化用药提供了理论依据.     

中药降脂灵片药效的因果分析

本文对一种新型中药降脂灵片的药效进行因果分析。实验数据样本量小且是混合变量类型,传统的统计方法难以处理,本文采用图模型的方法建立一个链图模型,直观地刻画了该药对反映机体抗氧化能力和血脂水平的4个指标的因果影响。     

速效感冒片中扑热息痛与咖啡因含量测定的一种新方法

扑热息痛和咖啡因是速效感冒片中的主要成分,其含量决定了速效感冒片的药效及副作用的大小.我们通过实验分别做出一定浓度扑热息痛溶液和咖啡因溶液的紫外吸光度随检测波长的变化曲线(λ-A曲线),并用M athem acica软件中曲线拟合的方法分别求出上述两条变化曲线的拟合方程,以这两个方程的线性组合形式拟合速效感冒片溶液紫外光谱的λ-A曲线,根据系数的比例关系求得扑热息痛和咖啡因在速效感冒片中的含量.简化了传统双波长法的操作,提高了检测速度.     

An alternative to the Pythagorean rule? Reevaluating Problem 1 of cuneiform tablet BM 34 568

The first problem of the Seleucid mathematical cuneiform tablet BM 34 568 calculates the diagonal of a rectangle from its sides without resorting to the Pythagorean rule. For this reason, it has been a source of discussion among specialists ever since its first publication, but so far no consensus in relation to its mathematical meaning has been attained. This paper presents two new interpretations of the scribe's procedure, based on the assumption that he was able to reduce the problem to a standard Mesopotamian question about reciprocal numbers. These new interpretations are then linked to interpretations of the Old Babylonian tablet Plimpton 322 and to the presence of Pythagorean triples in the contexts of Old Babylonian and Hellenistic mathematics.     

La tablette babylonienne AO 17264 du musée du louvre et le problème des six frères

This tablet contains only one problem, that of dividing a Trapezium in six parts, equal two by two, and determined by the length of their sides. The solution is given in terms of nineteen integers. How was this solution found? Analysis of the text shows that the initial trichotomy of the trapezium was produced by an arbitary calculation, that the calculator probably tried, without success, to provide a minimal solution, and that pythagorean triplets were central. We thus have instructive evidence about the ambitions and limits of Babylonian mathematics.     

Image data analysis and classification in marketing

Nowadays, the diffusion of smartphones, tablet computers, and other multipurpose equipment with high-speed Internet access makes new data types available for data analysis and classification in marketing. So, e.g., it is now possible to collect images/snaps, music, or videos instead of ratings. With appropriate algorithms and software at hand, a marketing researcher could simply group or classify respondents according to the content of uploaded images/snaps, music, or videos. However, appropriate algorithms and software are sparsely known in marketing research up to now. The paper tries to close this gap. Algorithms and software from computer science are presented, adapted and applied to data analysis and classification in marketing. The new SPSS-like software package IMADAC is introduced.     

Methods and traditions of Babylonian mathematics: Plimpton 322, Pythagorean Triples,and the Babylonian Triangle Parameter Equations

The remarkable Old Babylonian clay tablet which is commonly called Plimpton 322 was published originally by Neugebauer and Sachs in their now-classical Mathematical Cuneiform Texts of 1945. It contains, in a table with three preserved columns, a list of values of three quantities, which in the present paper are referred to as $\text{c}{}^2$, b, and c. It is easy to verify that the listed values (expressed in the usual Babylonian sexagesimal notation) are precisely the ones that can be obtained by use of the triangle parameter equations

$\text{b = a}\text{b}\text{, c = a}\text{c}\text{;}\text{b}\text{=}\text{1}\text{2}\text{(t′ ? t),}\text{c}\text{=}\text{1}\text{2}\text{(t′ ? t)}$

, if one allows the parameter t (with the reciprocal number $\text{t′ =}\text{1}\text{t}$ to vary over a conveniently chosen set of 15 rational numbers $\text{t =}\text{s}\text{r}$, and if the multiplier a is chosen in such a way that b and c become integers with no common prime factors. Hence, for every pair (b, c) appearing in the second and third columns of Plimpton 322, the corresponding triple (a, b, c) is a positive primitive Pythagorean triple, i.e., the coprime integers a, b, and c are the sides of a right triangle and therefore a solution of the indeterminate equation a² + b² = c², the so-called Pythagorean equation). After its publication by Neugebauer and Sachs the Plimpton tablet was further discussed and interpreted by a number of other authors (Bruins, Price, et al.) from several different points of view. It is the purpose of the present paper to try to extract and extend the best ideas from these various discussions and interpretations in order to achieve a unified and comprehensive analysis of the construction and meaning of this unique and important Babylonian mathematical text. In the paper a few comparisons with related texts are also made, for the purpose of showing that the table on Plimpton 322 is intimately associated with several other interesting aspects of Babylonian mathematics.     

Neither Sherlock Holmes nor Babylon: A Reassessment of Plimpton 322

Ancient mathematical texts and artefacts, if we are to understand them fully, must be viewed in the light of their mathematico-historical context, and not treated as artificial, self-contained creations in the style of detective stories. I take as a dramatic case study the famous cuneiform tablet Plimpton 322. I show that the popular view of it as some sort of trigonometric table cannot be correct, given what is now known of the concept of angle in the Old Babylonian period. Neither is the equally widespread theory of generating functions likely to be correct. I provide supporting evidence in a strong theoretical framework for an alternative interpretation, first published half a century ago in a different guise. I recast it using regular reciprocal pairs, Høyrup's analysis of contemporaneous “naı̈ve geometry,” and a new reading of the table's headings. In contextualising Plimpton 322 (and perhaps thereby knocking it off its pedestal), I argue that cuneiform culture produced many dozens, if not hundreds, of other mathematical texts which are equally worthy of the modern mathematical community's attention.Wir müssen frühe mathematische Texte und Objekte im Hinblick auf ihre mathematisch-historische Umgebung betrachten und sie nicht als künstliche, vollständige Schöpfungen im Stile von Detektivgeschichten behandeln, wollen wir sie verstehen. Als dramatische Fallstudies dient mir die Keilschrifttafel Plimpton 322. Ich zeige auf, dass die weitverbreitete Ansicht, so etwas wie eine trigonometrische Tabelle vor uns zu haben, nicht richtig sein kann, und zwar aufgrund unseres Wissens über die Vorstellung des Winkels in altbabylonischer Zeit. In gleiche Weise ist die gängige Theorie über erzeugende Funktionen wahrscheinlich falsch. Ich kann meine Neuinterpretation, die in einen stark theoretischen Rahmen eingebettet wird, mit Texten belegen. Hinter meiner Neuinterpretation liegt eine fünfzigjährige Theorie, die auf Bruins zurückgeht. Sie fundiert auf den Gebrauch von regelmassigen, reziproken Paaren, auf Høyrups Analyse der naiven Geometrie und auf eine neue Lesung der Überschriften der Tabelle. Indem ich die Keilschrifttafel Plimpton 322 in ihren historischen Kontext stelle, plädiere ich dafür, dass viele andere mathematische Texte mesopotamischen Ursprungs es ebenso verdienen, von uns beachtet zu werden. Copyright 2001 Academic Press.AMS Subject Classifications: 01A17; 01A85.