逻辑推理是六大核心素养之一,数学证明过程中所涉及的循环论证与学生的逻辑推理等核心素养有着很强的相关性。在实际教学中,循环论证并不少见。研究结合人教版普通教科书2019版(以下简称人教版)及各类练习题,按照知识点归纳高中数学学习中常见的循环论证问题,分为函数单调性、勾股定理、三角函数和立体几何四部分,为教材改革和教师教学提供新的方向。Logical reasoning is one of the six core literacy, and the circular argument involved in the mathematical proof process has a strong correlation with the core literacy of students’ logical reasoning. In actual teaching, circular argumentation is not uncommon. The research combines the 2019 edition of the ordinary textbook of the People’s Education Edition (hereinafter referred to as the People’s Education Edition) and various exercises to summarise the common circular argumentation problems in high school mathematics learning according to the knowledge points, which are divided into four parts: function monotony, Pythagorean theorem, trigonometric function and three-dimensional geometry, providing a new direction for textbook reform and teacher teaching.
数学教学在让学生掌握基本的知识技能的同时,还要促进学生数学核心素养的提升及水平的达成。“学习进阶”和“学历案”为培养数学核心素养提供了设计思路和操作框架。笔者以人教A版高中数学必修一的“函数的概念与性质”为例,结合将学习进阶融入学历案设计的三个要点,即要明确阶段性目标、要实现教学评一体化、要创设合适的教学情境,展示了融合学习进阶理论的完整单元学历案和课时学历案学习过程设计,分析了学习进阶理论与学历案设计结合的实践意义,以期为一线教师们提供参考。Mathematics instruction aims to not only equip students with basic knowledge and skills, but also promote the development of their mathematical core competencies and the attainment of their proficiency level. “Learning progression” and “Education program” provide design ideas and operational framework for cultivating mathematics core literacy. Taking “The Concept and Nature of Function” as an example, the author combined the three key points of integrating learning progression into the design of degree plan, that is, to clarify the stage goal, to realize the integration of teaching evaluation, and to create a suitable teaching situation, and demonstrated the design of the learning process of the complete unit degree plan and the class hour degree plan integrating the theory of learning progression. This paper analyzes the practical significance of the combination of learning progression theory and academic record design, in order to provide reference for teachers.
建构主义理论倡导以学生为核心,运用多样化的教学策略激发学生的知识自主建构与深刻理解。该研究选取“对数函数的图象和性质”为例,构建了基于建构主义理论的高中数学教学设计,通过反思和总结,旨在协助学生牢固掌握核心知识点,并锻炼其数学逻辑思维与问题解决技巧。Constructivist theory advocates for a student-centered approach, employing a variety of teaching strategies to stimulate students’ autonomous construction of knowledge and profound understanding. This study selects “the graph and characteristics of logarithmic functions” as an example, constructing a high school mathematics teaching design based on constructivist theory. Through reflection and summary, the aim is to assist students in firmly grasping core knowledge points and to exercise their mathematical logical thinking and problem-solving skills.
APOS理论是一种数学教育理论,强调通过操作、过程、对象和图式四个阶段促进学生的数学概念建构。余弦定理作为高中数学的核心内容,具有深刻的数学内涵和丰富的物理背景,是数形结合的良好载体和有效的解题工具。本研究以高中数学“余弦定理”相关内容为研究对象,采用文献研究法和案例分析法对APOS理论的来源、内涵与特点进行分析,结合余弦定理的教材内容与《普通高中数学课程标准(2017版)》解读,分析以APOS理论为指导的高中数学教学案例,提出余弦定理的教学策略和具体的教学设计案例。APOS theory is a kind of mathematics education theory, which emphasizes the promotion of students’ mathematical concept construction through the four stages of operation, process, object and schema. As the core content of high school mathematics, cosine theorem has profound mathematical connotation and rich physical background. It is a good carrier and effective problem-solving tool for the combination of number and shape. This study takes the relevant content of high school mathematics “cosine theorem” as the research object, uses literature research method and case analysis method to analyze the source, connotation and characteristics of APOS theory, combines the textbook content of cosine theorem with the interpretation of “Ordinary High School Mathematics Curriculum Standard (2017 Edition)”, analyzes the high school mathematics teaching cases guided by APOS theory, and puts forward the teaching strategies and specific teaching design cases of cosine theorem.
本文基于APOS理论,以初中数学中的分式概念为例,对数学概念教学进行了深入探究。文章首先系统梳理了APOS理论的核心要点,并指出了当前数学概念教学中存在的一些普遍问题。随后,结合分式的教学实际,详细分析了学生在掌握分式概念过程中可能经历的四个阶段,包括活动阶段(Action)、过程阶段(Process)、对象阶段(Object)和图式阶段(Schema),从而揭示分式概念形成和发展的内在规律,并设计相应教学过程。依据所设计的教学过程进行教学实践,课后通过调查问卷与测试成绩分析调查学生分式概念学习情况,数据分析表明,基于APOS理论的分式概念教学过程能达到较好的效果。通过这一探究,本文旨在帮助教师更好地理解分式概念的教学过程,把握学生在不同阶段的学习特点和难点,从而采取针对性的教学策略,提升教学质量。同时,引导学生更有效地掌握分式概念,提高数学学习的效果。This paper conducts an in-depth exploration of mathematics concept teaching, taking the concept of fractions in junior high school mathematics as an example, based on the APOS theory. It first systematically outlines the core points of the APOS theory and points out some common issues in current mathematics concept teaching. Subsequently, combining the practical teaching of fractions, the paper analyzes in detail the four stages that students may experience in mastering the concept of fractions, including the Action stage, Process stage, Object stage, and Schema stage. This analysis reveals the internal laws governing the formation and development of the fraction concept and designs a corresponding teaching process. Teaching practice is carried out according to the designed teaching process, and a post-class survey is conducted to investigate students’ learning of the fraction concept through questionnaires and test score analysis. Data analysis shows that the fraction concept teaching process based on the APOS theory can achieve good results. Through this exploration, this paper aims to help teachers better understand the teaching process of the fraction concept, grasp the learning characteristics and difficulties of students at different stages, and thereby adopt targeted teaching strategies to improve teaching quality. At the same time, it guides students to more effectively grasp the fraction concept and enhance the effectiveness of mathematics learning.
教育数字化的发展要求对教师的数字化能力提升有迫切的需求,教师的数字化意识很大程度上影响了教师的数字化能力。基于NVivo数据分析软件和扎根理论,对访谈文本资料进行三级编码,提炼出数字化意识的三个主范畴:数字化认识、数字化意愿和数字化意志,并构建职前教师数字化意识影响机理的理论模型。在此基础上提出教师数字化意识培育策略:整合数字化工具与数学教学,提高数字化认识;开展数字化教学实践活动,强化数字化意愿;构建数字化学习共同体。The development of digitalization in education demands an urgent need to enhance teachers’ digital competencies. Teachers’ digital awareness significantly influences their digital capabilities. Based on the NVivo data analysis software and grounded theory, three levels of coding were conducted on the interview text data, identifying three main categories of digital awareness: digital understanding, digital willingness, and digital volition. A theoretical model of the mechanisms influencing the digital awareness of pre-service teachers was constructed. Based on this, strategies for cultivating teachers’ digital awareness are proposed: integrating digital tools with mathematics teaching to improve digital understanding;engaging in digital teaching practice activities to strengthen digital willingness;and building a digital learning community.
文章探讨了波利亚解题理论在初中几何教学中的应用。波利亚解题理论包括弄清问题、拟定计划、执行计划、回顾反思四个步骤,为初中几何教学提供了系统方法和思路。在理论层面,该理论源于波利亚对数学教育的深入研究,在全球范围内广泛传播并产生深远影响。在实践应用方面,其在初中几何证明题发挥重要作用。如在证明题中利用等腰三角形简化问题、应用定义进行转化;在教学效果上,显著提升了学生的思维能力,培养了逻辑推理能力和创新思维,同时提高了教学质量,增强了学生学习兴趣。波利亚解题理论在初中几何教学中具有重要应用价值,为提高学生几何解题能力和培养数学素养提供有力支持。This paper discusses the application of Polya’s problem-solving theory in junior high school geometry teaching. Polya’s problem-solving theory includes four steps: clarifying the problem, formulating the plan, implementing the plan, and reviewing and reflecting, which provides a systematic method and ideas for junior high school geometry teaching. At the theoretical level, the theory stems from Polya’s in-depth research on mathematics education, which has been widely disseminated and has had a profound impact on the world. In terms of practical application, it plays an important role in junior high school geometry proof problems. For example, in the proof question, the isosceles triangle is used to simplify the problem and apply the definition for transformation;in terms of teaching effect, it has significantly improved students’ thinking ability, cultivated logical reasoning ability and innovative thinking, improved teaching quality, and enhanced students’ interest in learning. Polya’s problem-solving theory has important application value in junior high school geometry teaching, which provides strong support for improving students’ geometric problem-solving ability and cultivating mathematical literacy.
课程思政是新时代高等教育落实立德树人根本任务的核心路径。文章从马克思主义哲学、社会主义核心价值观与专业知识的融合视角出发,系统论证《概率论与数理统计》课程中融入思政教育的必要性与可行性。通过构建“一个核心、两个维度、三个层次”的理论框架,提出从学科发展史、核心知识点与社会热点三个路径挖掘思政元素,并设计案例教学、问题探究、实践项目等四类融入模式。教学实践表明,该模式能显著提升学生的专业认同度与社会责任感,为理工科课程思政建设提供可复制的范式。Curriculum ideological and political education is the core path for higher education in the new era to fulfill the fundamental task of establishing morality and cultivating people. This paper starts from the integration perspective of Marxist philosophy, socialist core values, and professional knowledge, and systematically demonstrates the necessity and feasibility of integrating ideological and political education into the “Probability Theory and Mathematical Statistics” course. By constructing a theoretical framework of “one core, two dimensions, and three levels”, it is proposed to excavate ideological and political elements from three paths: the disciplinary development history, core knowledge points, and social hotspots, and to design four types of integration modes: case teaching, problem exploration, practical projects, and others. Teaching practice shows that this mode can significantly improve students’ professional identity and social responsibility, providing a replicable paradigm for the ideological and political construction of science and engineering courses.
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