浅谈合情推理的教学

论述了合情推理与逻辑推理的关系，在高等数学教学中渗透合情推理的意义，并结合教学实践就通过合情推理引入数学理论，探求问题结论，探索解题方法作了初步探讨。     

新课程理念下的合情推理教学

传统数学教学中,教师常常过分强调“三段论”的演绎推理,忽视合情推理．新课程倡导让学生“经历观察、实验、猜想、证明等数学活动,发展合情推理能力和初步的演绎推理能力”,就是要在数学教学中,引导学生进行探索,发现问题、解决问题,体验数学发展的过程,学会用归纳、类比和演绎进行推理．如今新的高中课程设置把合情推理纳入到选修教材中,无疑把合情推理教学推向了高潮．     

浅谈数学中的合情推理

合情推理是数学发现的重要方法,本文介绍了合情推理的概念、作用、途径.所论及的内容对数学教学有参考价值.     

“演绎与合情”融合的推理核心素养评价探析——以2023年成都中考数学压轴题为例

推理素养是重要的数学核心素养之一,推理一般包括合情推理和演绎推理,合情推理又可分为归纳推理和类比推理.2023年成都中考数学压轴题通过创设不同情境,探索几何推理试题的特征,融合考查合情推理和演绎推理,本文先对该试题进行解析,然后给出教学启示:教师日常教学中要重视培育学生推理核心素养发展的适切性、层递性和协调性.     

人教A版“推理与证明”若干教学问题的诠释

问题1为什么要将《合情推理》写进课标教材？ 长期以来，大纲教材没呈现《合情推理》，高中数学教学比较注重发展学生的演绎推理能力，忽视了合情推理，课标教材中单独提出这一内容，那是由于合情推理有发现新结论、探索解决问题的思路和方法的作用，是一种具有创造性的思维活动．随着国内外数学教育的发展，越来越认同在课堂教学中对合情推理能力培养的重要性与必要性．     

泰勒级数的发现及其思想方法

研究了泰勒级数的发现过程,分析了其中的数学思想方法,指出了数学思想的继承性以及合情推理在数学发现中的作用.在数学教学中应重视合情推理的教学,它是培养学生创新能力的有效手段.     

合情推理——启航学生思维小船的舵手

合情推理有"归纳"和"类比"两种推理模式,这种推理是建立在观察、实验的基础上,通过"类比"来产生"联想",或者通过"归纳"来进行"猜想",是一种"发现未知"的思维形式.在新课教学中,如能针对具体问题,恰当运用"合情推理",可助推启航学生思维的小船,让学生的创造性思维畅游在知识的海洋中,较好地完成课堂学习之旅.这样的新课教学,有利于开发学生的创造性素质,对于培养创造性人才有着积极意义.本文基于"余弦定理"的新课教学,以引导探究为教学手段,运用合情推理,从余弦定理的生成方式这个角度进行案例剖析.关注培养学生合情推理意识的新课教学,往往可以取得出人意料的教学效果.     

合情的推理错误的结果——一道高考试题的错因分析

新课标把合情推理列入到选修2—2的数学内容中,所谓合情推理,就是根据已有的事实,经过观察、分析、比较、联想,再进行归纳、类比,然后提出猜想的推理,我们把它统称为合情推理,通俗地说,合情推理是指“合乎情理”的推理,在数学研究中,得到一个新结论之前,合情推理常常能帮助.     

论大学数学教学中的合情推理

本文讨论合情推理的含义及其与论证推理的关系 ,和它在大学数学教学中的意义 .     

新课程中“类比推理”的教学感悟及典型案例研究

《数学课程标准》中新增加“推理与证明”包含演绎推理与合情推理,新一轮基础教育数学课程改革中,给了合情推理应有的关注．《数学课程标准》在选修1—2与选修2—2中设计了推理与证明内容,要求学生结合已学过的数学实例和生活的实例,对合情推理与演绎推理的方法进行概括总结,体会合情推理与演绎推理在数学结论发现与数学体系建构中的作用．而类比作为一种常用的合情推理方法,具有猜测和发现结论、探索和提供思路的作用,有利于创新能力的培养．本文结合高考试题实例,从概念类比、方法类比、升维类比、结构类比四个角度,对近几年高考试卷中出现的“类比”型试题进行分类解析,探讨教学实践中对学生类比推理能力的培养．     

刍议大学数学教学活动的问题与对策

当前各高校的大学数学教学或多或少存在一些不协调的地方,如理科类学生课程深度过大,文科类学生数学能力弱,考核试卷的难度控制不好等等.针对大学数学的教学法及大众化教育的分析研究,结合教学实践,提出大学数学教学改革的一些对策.     

新课标体系下高中数学对大学工科数学教学产生的问题分析及对策探索

大学工科数学的部分基础内容下移于高中数学,高中数学中原有的基础内容又有所删减,大学工科数学的教材内容并没有因此而调整.所以,新课标体系下的高中数学教学对大学工科数学教学产生了深刻的影响.本文分析了大学工科数学教学面临的问题,在此基础上,结合作者多年的教学实践,探索性的提出了解决问题的对策.     

Measurement of Length: The Need for a Better Approach to Teaching

Three hundred and eighty-three children in grades 1–5 were individually interviewed to find out at when they construct unit iteration out of transitive reasoning as described by Piaget, Inhelder, and Szeminska (1960) . The results indicated that most children (72%) construct transitive reasoning by second grade and that most (76%) construct unit iteration out of transitive reasoning by fourth grade. The article explains why traditional instruction produces the poor results revealed by the National Assessment of Educational Progress. It also suggests a better approach to the teaching of measurement that presents problems and encourages children to modify their ways of thinking.     

模糊集的相对扰动

在模糊推理的理论研究中,模糊推理方法的扰动性分析是一个重要的研究方向,即讨论模糊推理前件的变化对后件的影响。本文给出模糊集的相对扰动的概念,讨论由模糊集运算产生的相对扰动,以及模糊推理的相对扰动性,给出一些有趣的性质。     

On recognizing proportionality: Does the ability to solve missing value proportional problems presuppose the conception of proportional reasoning?

This paper investigates the relationship between the ability of middle school students to solve missing value proportional problems and their facility in differentiating proportional relationships from non-proportional relationships. Students in low- and high-proficiency groups in mathematics took a ratio-and-proportion test involving two typical missing value proportional (MVP) and two recognizing proportionality (RP) problems. The findings revealed that while the students generally performed better on MVP problems than on RP problems, the two groups differed in their performance on MVP problems, but not on RP problems. Moreover, of those students from both the groups who successfully solved the two MVP problems, a significantly greater proportion of students in the high-proficiency group were unsuccessful in solving either of the two RP problems than those in the low-proficiency group. An analysis of performance differences between items within the same student group showed that the effect of differences in the structural components of RP problems to some extent contradicted the previous findings on the effect of differences in the structural components of MVP problems. It is hoped that these findings can shed light on what might be missing in the teaching and learning of proportional reasoning.     

Learning mathematics through algorithmic and creative reasoning

There are extensive concerns pertaining to the idea that students do not develop sufficient mathematical competence. This problem is at least partially related to the teaching of procedure-based learning. Although better teaching methods are proposed, there are very limited research insights as to why some methods work better than others, and the conditions under which these methods are applied. The present paper evaluates a model based on students’ own creation of knowledge, denoted creative mathematically founded reasoning (CMR), and compare this to a procedure-based model of teaching that is similar to what is commonly found in schools, denoted algorithmic reasoning (AR). In the present study, CMR was found to outperform AR. It was also found cognitive proficiency was significantly associated to test task performance. However the analysis also showed that the effect was more pronounced for the AR group.     

“Playing the game” of story problems: Coordinating situation-based reasoning with algebraic representation

This study critically examines a key justification used by educational stakeholders for placing mathematics in context –the idea that contextualization provides students with access to mathematical ideas. We present interviews of 24 ninth grade students from a low-performing urban school solving algebra story problems, some of which were personalized to their experiences. Using a situated cognition framework, we discuss how students use informal strategies and situational knowledge when solving story problems, as well how they engage in non-coordinative reasoning where situation-based reasoning is disconnected from symbol-based reasoning and other problem-solving actions. Results suggest that if contextualization is going to provide students with access to algebraic ideas, supports need to be put in place for students to make connections between formal algebraic representation, informal arithmetic-based reasoning, and situational knowledge.     

高等数学创造性思维教学的策略优化

论述了高等数学创造性思维教学的策略优化问题,提出了作者设计的高等数学创造性思维教学链,并结合高等数学的具体教学实践进行了多方位的探讨.     

将数学建模融入高等数学教材

根据新编教材编写体会,简述了将数学建模的思想与方法融入到高等数学内容中的尝试.     

Abduction-induction (generalization) processes of elementary majors on figural patterns in algebra

The article deals with issues concerning the abductive-inductive reasoning of 42 preservice elementary majors on patterns that consist of figural and numerical cues. We discuss: ways in which the participants develop generalizations about classes of abstract objects; abductive processes they exhibit which support their induction leading to a generalization; ways they justify their generalizations in the abductive stage, and; the effects of figural and numerical cues in the manner they construct a plausible abductive generalization. Two types of abductions are explored, model-based and manipulative. A proposed abductive-inductive reasoning process for pattern sequences is presented and discussed in the concluding section.