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| 类比与联想在数学教学中的应用
摘要
类比与联想思想是一种既实用又重要的中学数学思想，不论是在教师授课环节还是学生学习过程都能产生一定影响。在数学发展过程中，许多数学家发现重要定理和公式的过程都用到了类比与联想，18世纪的伟大数学家和物理学家欧拉就曾经用归纳与类比做出了很多著名的数学发现。学生掌握类比与联想思想并运用自如对于学习与理解新知识、总结知识内容都有重要意义，可以明显增强学习效果。本文旨在阐述类比与联想在中学数学教学过程中的应用，因为中学数学知识有很多用到类比与联想的思想，而且根据数学新课程标准的修订，在教学过程中“类比与归纳”占有较高地位，所以我们应给予重视。
本文从以下四个方面进行研究：
第一个方面是类比与联想在数学教学中的理论研究。本文从教育期刊以及教材知识入手，解读类比与联想的概念，再结合中学教学过程中有关类比与联想的教学设计思路以及教学设计原则，对这种数学思想进行理论方面的研究。其中教学设计思路分为三个步骤，分别是准备过程，实施过程以及验证过程，教学设计原则分为：共同参与原则，目标导向性原则以及过程性原则。
第二个方面是类比与联想在数学教学中的实例研究。利用教育实习经验，通过实例研究课堂上如何运用类比与联想进行教学及二者对数学教学的作用。本文根据类比与联想两方面的用途：引出新知识和总结知识内容，分别列举了导数的概念和三角函数的图像和性质两课。
第三个方面是类比与联想在数学解题中的应用。根据类比的分类四个角度来分析各种试题，分别有：概念类比型、方法类比型、升维类比型及结构类比型，从而感受与体会类比与联想在解题中的重要应用，为一线教师的课堂提供一定有价值的参考。
第四个方面是教师在实际教学中应如何培养学生的类比联想能力。就学生的学情基础提出创新的可行性建议，为学生提供思维时空，最大发挥教学成效。本文就建议方面提出三大点：课堂认识类比含义，展现类比巧妙作用以及寻觅知识点间的类比。
最后对本文研究内容的创新之处与不足之处进行总结与分析，希望本文对于一线中学教师具备一定参考价值。
关键词：类比与联想思想；中学数学
Abstract
The thought of analogy and association is a very important thought in middle school mathematics, which is very practical both in the teaching process of teachers and in the learning process of students. In the development of mathematics, many mathematicians used analogy and association in discovering important theorems and formulas. Euler was a great mathematician and physicist in the 18th century, and he made many famous mathematical discoveries by induction and analogy. It is of great significance for students to master the ideas of analogy and association and use them freely for learning and understanding new knowledge and summarizing knowledge content, which can obviously enhance the learning effect. This paper aims to explain the application of analogy and association in middle school mathematics teaching process, because there are many ideas of analogy and association in middle school mathematics knowledge, and according to the revision of new mathematics curriculum standards, "analogy and induction" can not be ignored in the teaching process.
This paper studies from the following four aspects:
The first aspect is the theoretical research of category and association in mathematics teaching. This paper starts with the knowledge of educational journals and teaching materials, interprets the concepts of category and association, and then makes a theoretical research on this mathematical thought by combining the instructional design ideas and principles of category and association in the middle school teaching process. Among them, instructional design ideas are divided into three steps, namely, preparation process, implementation process and verification process. Instructional design principles are divided into: participation principle, goal-oriented principle and process principle.
The second aspect is the case study of category and association in mathematics teaching. Using the experience of educational practice, this paper studies how to use analogy and association in classroom teaching and their effects on mathematics teaching through examples. According to the uses of analogy and association: eliciting new knowledge and summarizing knowledge content, this paper lists the teaching design and reflection of the concept of derivative, the teaching design and reflection of the image and nature of trigonometric function.
The third aspect is the application of category and association in solving mathematical problems. According to the classification of analogy, this paper analyzes various test questions, including conceptual analogy, methodological analogy, dimensional analogy and structural analogy, so as to realize the application of analogy and association in solving problems and provide some reference for front-line teachers.
The fourth aspect is how to cultivate students' ability of analogy and association. Put forward innovative feasible suggestions on the basis of students' learning situation, provide students with thinking time and space, and give full play to teaching effectiveness. This paper puts forward three suggestions: understanding the meaning of analogy in class, showing the clever function of analogy and finding the analogy between knowledge points.
Finally, this paper summarizes and analyzes the innovations and shortcomings of this research content, hoping that this paper has certain reference value for teachers in first-line middle schools.
Key words:Analogy and association thought;Middle school mathematics
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