分类讨论

分类讨论是数学中的一种重要的思想方法和解题策略 .当问题所给的对象不宜进行统一的研究 ,或推理只有用分组的形式才能方便地表示出来时 ,就需要对研究的对象进行分类后对每一类分别研究 ,得出每一类的结果 ,最后综合各类的结果得到整个问题的结果 ,它是逻辑划分思想在解数学题中的具体运用 ,这种数学思想方法几乎涉及中学数学内容的各个部分 ,它可以把整体化为局部 ,把复杂问题化为单一问题 ,以便于“各个击破” .　分类讨论往往产生在思维受阻或不畅的时候 ,选择好的思维着眼点是使思维过程顺利发展的关键 ,也是认识为什么要分类讨论 ,以…     

例谈回避分类讨论的优化策略

分类讨论是一种重要的数学思想方法和解题策略，渗透到整个中学数学每个章节，由于这类题目综合性强，逻辑性严，探索性开放，自然也是难点．我们在重视分类讨论思想应用的基础上，也要注意克服动辄加以讨论的思维定势，要充分挖掘数学问题中潜在的特殊性，尽力打破常规，避免不必要的分类讨论．下面通过举例谈谈如何避免分类讨论的优化策略．     

数学解题中的分类讨论

数学中的许多问题需要分类讨论，做这类题需明确两点：（1）为什么要分类讨论，（2）分类讨论的标准．     

分类讨论思想在初中数学解题教学中的运用探究

在初中数学教学中,培养学生掌握分类讨论的能力可以提高学生的解题速度,并且在推进初中教学课程改革创新的大环境下,初中阶段数学测试题目也更注重检测学生掌握分类讨论的程度.所以,教师在课堂上要多培养学生分类讨论、从多个方面分析、独立思考等能力,本文重点探究的是如何应用分类思想解答初中数学题目.     

避免分类讨论,简化有关参数问题

数学中有许多问题涉及参数 ,分类讨论是一种重要的解题策略 ,有关的刊物也刊登过如何进行分类讨论解决含有参数问题的文章 ,但本人在长期的教学中发现有相当数量的看似需要分类讨论来解决的含参数问题 ,可以避免分类讨论 ,从而优化解题过程 .1　消去参数 ,避免分类讨论例 1　设 0 <ｘ <1 ,ａ >0且ａ≠ 1 ,比较|ｌｏｇａ( 1 -ｘ) |与 |ｌｏｇａ( 1 ｘ) |的大小 .解　 ｌｏｇａ( 1 -ｘ)ｌｏｇａ( 1 ｘ) =ｌｏｇ( 1 ｘ) ( 1-ｘ) =ｌｏｇ( 1 ｘ)( 1-ｘ) ( 1 ｘ)( 1 ｘ) =ｌｏｇ( 1 ｘ) ( 1-ｘ2 ) - 1 .∵ 0 <ｘ <1 ,∴ 1 <1 ｘ <2 ,0 <1 -ｘ2…     

从一道高考题的解答谈分类讨论思想

2005年江苏高考卷的第22题是一道以研究函数性质为载体,重点考查学生分类讨论思想方法掌握情况的经典题目.题目难度不算大,从考后与学生的交流当中得知,许多学生由于弄不清解答过程中的错综复杂的分类讨论而中断解题思路.究其原因,是由于对分类讨论思想方法掌握的不透彻造成的.     

我们是如何陷入分类讨论的？

分类讨论是一种重要的数学思想和解题策略,在中学数学学习中有重要的位置．当然,由于分类讨论,也难免使得问题的解决过程变得繁杂冗长．因此,我们又希望避免解题过程中的分类讨论．事实上,解决某些数学问题,之所以要分类讨论,常常是囿于我们所选择的解题视角,而不是问题本身的缘故．     

如何在解题中运用分类讨论思想

数学思想是数学知识、数学技能、数学方法的本质体现,是形成数学能力,数学意识的桥梁.因而在《课标》中,数学思想被视为数学基础的重要组成部分,而分类讨论思想是十分重要的数学思想. 分类讨论思想逻辑性强,它不仅用于数学解题,而且在其他领域也有广泛的应用.通过数学中的分类解题,可以增强分类的意识,拓宽解题的空间,培养全面解决问题的能力. 近年来,在中考或数学竞赛中,经常出现多解问题,不少学生往往不注意这一点,很容易导致漏解,使答案不完整.为了保证求得的答案正确、合理,应正确应用分类思想指导解题.     

避免"分类讨论"的几种策略

分类讨论既是一种重要的数学思想方法,又是一种重要的解题策略,在数学解题中有着广泛的应用.但分类讨论时,一般过程都较为冗长、繁琐,且极易在完备性上造成失误.因此,在分类之前应有意识地调整思维策略,尽量地避免分类讨论,以简化或优化解题过程,达到简捷解题的目的.本文介绍几种避免分类讨论的解题策略.……     

Solution of Poisson's equation with global,local and nonlocal boundary conditions

Boundary value problems (BVPs) for partial differential equations are common in mathematical physics. The differential equation is often considered in simple and symmetric regions, such as a circle, cube, cylinder, etc., with global and separable boundary conditions. In this paper and other works of the authors, a general method is used for the investigation of BVPs which is more powerful than existing methods, so that BVPs investigated by the method can be considered in anti-symmetric and arbitrary regions surrounded by smooth curves and surfaces. Moreover boundary conditions can be local, non-local and global. The BVP is expanded in a convex and bounded region D in a plane. First, by generalized solution of the adjoint of the Poisson equation, the necessary boundary conditions are obtained. The BVP is then reduced to the second kind of Fredholm integral equation with regularized singularities.     

数学学习与数学思想方法

数学学习,实际上就是对数学知识的理解和对数学思想方法的掌握与运用,数学思想方法是对数学知识的概括,也是数学知识的本质所在.对数学思想方法进行层次性划分,使数学学习具有针对性,同时也从方法论角度提供了数学学习的方法.全面地掌握数学方法,不仅有助于对数学知识的理解和运用,能有效提高数学的学习效率,对提高个体的整体素养也具有重要的现实意义.     

杰出人物论微积分

摘编14位杰出人物对微积分的精辟论述。     

The movement towards a more experimental approach to problem solving in mathematics using coding

Motivated by a problem proposed in a coding competition for secondary students, I will show on this paper how coding substantially changed the problem-solving process towards a more experimental approach.     

突出数学建模思想培养学生创新能力

论述了在高等数学教学中突出数学建模思想对培养学生创新能力的意义,提出了在教学中的一些主要环节突出数学建模思想的方法.     

恒等变形与高等数学创意解题

高等数学解题中常用恒等变形.对常用恒等变形类型与方法技巧进行归纳整理并在教学中加以应用,有助于提高学生解题能力和效率.领悟其“恒”与“变”转换,对培养学生的创新创意思维有积极的影响.     

高等数学在中学数学建模中指导作用

本文用典型的例子说明如何在中学数学建模中有效地使用高等数学的思想和方法 ,促进高等数学和中学数学的有机结合 .     

Students’ Activity in Computer-Supported Collaborative Problem Solving in Mathematics

The purpose of this study was to analyse secondary school students’ (N = 16) computer-supported collaborative mathematical problem solving. The problem addressed in the study was: What kinds of metacognitive processes appear during computer-supported collaborative learning in mathematics? Another aim of the study was to consider the applicability of networked learning in mathematics. The network-based learning environment Knowledge Forum (KF) was used to support students’ collaborative problem solving. The data consist of 188 posted computer notes, portfolio material such as notebooks, and observations. The computer notes were analysed through three stages of qualitative content analysis. The three stages were content analysis of computer notesin mathematical problem solving, content analysis of mathematical problem solving activity and content analysis of the students’ metacognitive activity. The results of the content analysis illustrate how networked discussions mediated mathematical knowledge and students’ questions, while the mathematical problem solving activity shows that the students co-regulate their thinking. The results of the content analysis of the students’ metacognitive activity revealed that the students use metacognitive knowledge and make metacognitive judgments and perform monitoring during networked discussions. In conclusion, the results of this study demonstrate that working with the networked technology contributes to the students’ use of their mathematical knowledge and stimulates them into making their thinking visible. The findings also show some metacognitive activity in the students’ computer-supported collaborative problem solving in mathematics.     

Exploring a structure for mathematics lessons that initiate learning by activating cognition on challenging tasks

While there is widespread agreement on the importance of incorporating problem solving and reasoning into mathematics classrooms, there is limited specific advice on how this can best happen. This is a report of an aspect of a project that is examining the opportunities and constraints in initiating learning by posing challenging mathematics tasks intended to prompt problem solving and reasoning to students, not only to activate their thinking but also to develop an orientation to persistence. Data were sought from teachers and students in middle primary classes (students aged 8–10 years) via online surveys. One lesson focusing on the concept of equivalence is described in detail although mention is made of other lessons. The research questions focused on the teachers’ reactions to the lesson structure and the specifics of the implementation in a particular school. The results indicate that student learning is facilitated by the particular lesson structure. This article reports on the implementation of this lesson structure and also on the finding that students’ responses to the lessons can be used to inform subsequent learning experiences.     

“We definitely wouldn't be able to solve it all by ourselves,but together…”: group synergy in tertiary students' problem-solving practices

The ability to address and solve problems in minimally familiar contexts is the core business of research mathematicians. Recent studies have identified key traits and techniques that individuals exhibit while problem solving, and revealed strategies and behaviours that are frequently invoked in the process. We studied advanced calculus students working in groups to identify what strategies they employed and how, including what encouraged the opportunities to invoke them. The study revealed behaviours not included in the original taxonomy, including one that we termed ‘group synergy’. We propose extensions to Carlson and Bloom's original taxonomy to encompass group behaviour and identify the importance of these behaviours in developing problem-solving skills. Finally, we suggest improvements for future problem-solving session iterations, with the goal of promoting opportunity for more expert performance.