数学理解三种方式及其课堂教学特征

数学理解有三种方式,即记忆性理解、解释性理解和探究性理解。其中,记忆性理解的教学只要求学生记住事实材料,通过机械记忆、模仿与简单套用,反复训练学生的记忆能力。解释性理解的教学通过教师对原理、理论的系统讲解发展学生的理解能力,但学生得到的仍是教师传授的内容,而不是学生自己的领悟。探究性理解的教学则是以问题为中心,引起学生对重要问题产生困惑,通过对话和交流引导学生独立探索发现规律和建构知识的意义。三种数学理解方式分别对应着行为主义、认知主义和建构主义的学习心理观。课堂教学应综合运用三种理解方式,力求让学生达到探究性理解。     

七年级学生数学理解水平的调查研究——以“科学记数法”为例

数学学习重在理解,但教学中发现学生对科学记数法的理解却处于较低的水平,这不利于学生未来相关学科学习.对此,基于“记忆性理解、解释性理解和探究性理解”的数学理解水平框架设计了相关测试问卷,调查七年级学生数学理解水平的达成情况.发现:学生对科学记数法的数学理解水平基本达到了记忆性理解水平,解释性理解水平处于中等位置,探究性理解水平的培养任重道远.对此,提出教师在教学中要重视数学理解的深度发展;多元视角下关注数学理解;发挥数学理解的价值意义.     

高中生对数学理解性学习认识的因素结构

高中生对数学理解性学习认识的3个因素为探究性理解因素、记忆性理解因素和解释性理解因素.重点班学生在记忆性理解因素上的平均得分显著地高于普通班学生的得分;重点班学生在解释性理解因素上的平均得分也显著地高于普通班学生的得分.在探究性理解因素上,重点班学生的平均得分也高于普通班学生的得分,但没有达到统计意义上的显著水平.数学理解性学习的各因素在性别变量上基本上不存在统计意义上的显著性差异.     

基于数学理解的“直线与圆的位置关系”教学实录与思考

正徐彦辉在《数学理解三种方式及其课堂教学特征》一文中指出,数学理解有三种方式,即记忆性理解、解释性理解和探究性理解。其中,记忆性理解的教学只要求学生记住事实材料,通过机械记忆模仿与简单套用;解释性理解的教学是通过教师对原理、理论的系统讲解,发展学生的理解能力,但学生得到的仍是教师传授的内容,而非自身领悟的     

初中生数学理解水平的测试调查研究

在"记忆水平、解释性理解水平和探究性理解水平"为研究数学理解的理论框架下,设计测试题,采取随机分层抽样的方式,调查初中学生3层次数学理解水平的达成情况.发现:记忆和解释性理解水平的目标己基本达成,探究性理解水平的达标远低于解释性理解水平,探究性理解能力的培养任重而道远.     

中学生数学理解性学习的现状调查与改进策略

以记忆性理解、解释性理解和探究性理解三个维度的量表式问卷为工具,调查温州市六所不同类型的学校中学生数学理解性学习的现状。研究发现:当前中学生由于受到性别、学段、学校类别和学校区位等因素的制约和影响,其数学理解方式在上述三个维度上表现出不同的特点和差异。由此提出:从改进学生数学理解性学习的方式、关注数学理解性学习方式的性别差异和优化教育资源的合理配置三个层面来提升中学生数学理解性学习的质量。     

数学观、数学理解方式与学业成绩关系研究

采用自编的数学观和数学理解方式量表式问卷对一所重点初中的学生进行调查,考查初中生的数学观、数学理解方式及其对数学学业成绩的影响.结果表明:(1)初中生的数学观、数学理解方式、数学学业成绩两两之间呈显著正相关,且数学理解方式的每一个维度都与数学学业成绩呈极其显著的正相关.(2)数学观的三个维度对数学学业成绩的影响不显著,数学理解方式的三个维度对数学学业成绩影响显著.数学理解方式三个维度可以预测和解释数学学业成绩变异量的8%,其中解释性理解对数学学业成绩的预测与解释力最大.(3)高、中成绩组学生的解释性理解得分显著高于低成绩组学生,高成绩组学生的记忆性理解得分显著高于中、低成绩组学生,而高、中成绩组学生与中、低成绩组学生之间的探究性理解差异则不显著.     

教师课堂教学行为对学生学习的影响

我国著名教育家陶行知先生曾说过：“我以为好的先生不是教书,不是教学生,乃是教学生学.”学生在课堂上的学习表现,除了与原有的认知基础有关外,更多的是与教师的教学行为有着密切的联系.课堂教学行为是指教师受一定的思想支配,在学校课堂这一专门的教学场所内进行的教学、组织管理活动等的表现.顾泠沅先生在《教学实验论》中总结上海青浦实验研究经验时指出：根据课堂的教与学的情况,教师的教学行为可以分为三种水平：记忆性水平的教学、解释性理解水平的教学以及探究性理解水平的教学.     

数学教学中学生认知水平下降的成因和对策

从认知角度看,教学水平一般可分为三个层次：记忆（记住事实,操作程序）、解释性理解（教师讲解,学生领会,简称解释水平）、探究性理解（学生投入,亲自探索,简称探究水平）。近年来,大量的课堂观察发现,由于教师的教学观念滞后,或片面追求教学活动的外在形式,掩盖了教学活动的深刻性,从而使数学教学中认知水平下降,违背了课程改革的初衷,显然不利于培养学生终身学习的能力。现笔者结合自己平时的观察、学习和反思,写成此文,以求教于同行。     

让概念学习由浅入深

概念教学是小学数学教学的重要内容,但很多教师的概念教学常常过于浅显,浮于表面,导致学生的学习效率不高。教师应从研读教材、基于学生、梳理脉络三方面出发,引导学生深入建构概念、理解概念和完善概念,从而让学生的概念学习由浅入深。     

关于奥数与小学生数学思维能力发展冲突的思考

近年来,小学“奥数”越来越热。小学生做奥数题是否真的有利于小学生的数学思维能力发展,这一问题成为一些教育界人士与家长关心的重要议题。文章认为,大面积的小学生“奥数”培训热,降低了小学生数学学习的兴趣,不利于提高小学生的数学能力。小学数学教学应该把提升小学生的数学素质作为基本出发点,促进小学生的数学思维能力发展。     

Characterizing pivotal teaching moments in beginning mathematics teachers’ practice

Although skilled mathematics teachers and teacher educators often “know” when interruptions in the flow of a lesson provide an opportunity to modify instruction to improve students’ mathematical understanding, others, particularly novice teachers, often fail to recognize or act on such moments. These pivotal teaching moments (PTMs), however, are key to instruction that builds on student thinking about mathematics. Video of beginning secondary school mathematics teachers’ instruction was analyzed to identify and characterize PTMs in mathematics lessons and to examine the relationships among the PTMs, the teachers’ decisions in response to them, and the likely impacts on student learning. These data were used to develop a preliminary framework for helping teachers learn to identify and respond to PTMs that occur during their instruction. The results of this exploratory study highlight the importance of teacher education preparing teachers to (a) understand the mathematical terrain their students are traversing, (b) notice high-leverage student mathematical thinking, and (c) productively act on that thinking. This preparation would improve beginning teachers’ abilities to act in ways that would increase their students’ mathematical understanding.     

中学生数学理解障碍及教学对策的探究

数学理解障碍是数学学习中不容忽视的一种障碍．从数学认知、教育心理学等层面来讲，中学生的数学理解障碍有认知型障碍，表象型障碍，联系型障碍，语言型障碍；克服这些障碍相应的教学对策有：帮助学生生成正确的数学表象；利用实物、模型等，增强学生对知识的感性认识；利用多媒体进行辅助教学；注重数学交流；加强学生对数学知识的自主探究与实际应用等来加深理解。     

‘Crossing’ the equals sign: insights into pre-service teachers’ understanding of linear equations

This paper reports on an exploratory study designed to determine and enhance the conceptual understanding of a group of pre-service mathematics teachers at one Irish university utilizing an established framework for understanding mathematics. 23 students on a one year Professional Diploma in Mathematics Education participated in the study, which involved the distribution of a pre- and post-test and engagement in a ten week intervention designed to enhance their subject matter knowledge (SMK) and pedagogical content knowledge (PCK). The findings highlight that although there was an improvement in overall conceptual understanding across the entire cohort at the end of the intervention, within certain mathematical topics there was a statistically insignificant improvement and many deep-rooted issues remain. In this paper we focus on the pre-service teachers’ understanding of elementary algebra, in particular, how to solve a linear equation.     

理解的“回归对应”逻辑及其课堂实现——基于诠释学的解读

作为人的自我重塑活动,理解始终是课堂教学的基础和质量表征。已有研究缺乏诠释学一般理解逻辑和课堂本性的把握,难以彰显课堂理解的本质。长时段历史考察及逻辑分析表明,诠释学转向催生了理解的"回归对应"逻辑:理解即理解文本创生的世界,世界间遵循回归原理,世界内恪守对应法则。课堂中的"回归对应",意味着人类优秀文化作品是理解的对象,教师不是理解者,而是学生理解的引导者,发展学生理解力是课堂的终极目标。"回归对应"的课堂实践,要把学生理解力的培育植于理解活动之中,提升教师引导理解的能力,注重学生理解方法的教导。     

Beyond Motivation: Exploring Mathematical Modeling as A Context for Deepening Students' Understandings of Curricular Mathematics

Views of mathematical modeling in empirical, expository, and curricular references typically capture a relationship between real-world phenomena and mathematical ideas from the perspective that competence in mathematical modeling is a clear goal of the mathematics curriculum. However, we work within a curricular context in which mathematical modeling is treated more as a venue for learning other mathematics than as an instructional goal in its own right. From this perspective, we are compelled to ask how learning of mathematics beyond modeling may occur as students generate and validate mathematical models. We consider a diagrammatic model of mathematical modeling as a process that allows us to identify how mathematical understandings may develop or surface while learners engage in modeling tasks. Through examples from prospective teachers' mathematical modeling work, we illustrate how our diagrammatic model serves as a tool to unpack the intricacies of students’ mathematical experience while engaging in modeling tasks.     

‘I’m not a “maths‐person”!’ Reconstituting mathematical subjectivities in aesthetic teaching practices

In this study I have investigated how alternative ways of teaching mathematics influence and affect Early Childhood Education (ECE) students’ attitudes towards maths and how they understand their own subjectivities as more or less mathematical during a 10‐week alternative maths course. The investigated course adopts a feminist post‐structural approach based on critical pedagogy and deconstructive theory and includes an interdisciplinary approach to investigative mathematics. The data used include the memory/narrative writings and process‐writings of 75 female teacher‐education students, collected from three different cohorts, in which the students describe their learning processes throughout the maths course. The study shows that, in the main, the students became much more positively inclined to the subject of mathematics after the maths course and agreed that this course had changed their understanding of their own mathematical subjectivity, albeit in different and varying ways.     

Mathematics for teaching and deep subject knowledge: voices of Mathematics Enhancement Course students in England

This article reports an investigation into how students of a mathematics course for prospective secondary mathematics teachers in England talk about the notion of ‘understanding mathematics in depth’, which was an explicit goal of the course. We interviewed eighteen students of the course. Through our social practice frame and in the light of a review of the literature on mathematical knowledge for teaching, we describe three themes that weave through the students’ talk: reasoning, connectedness and being mathematical. We argue that these themes illuminate privileged messages in the course, as well as the boundary and relationship between mathematical and pedagogic content knowledge in secondary mathematics teacher education practice.     

Analyzing the effects of mathematical discourse-based instruction on eleventh-grade students’ procedural and conceptual understanding of probability and statistics

This study utilized discourse-based instruction as an alternative method of instruction that emphasizes the teaching of mathematics by actively engaging students in mathematical discourse practices. A quasi-experimental study was employed to determine the effectiveness of mathematical discourse-based instruction in enhancing eleventh-grade students’ conceptual and procedural understanding of probability and statistics. A researcher-constructed test instrument was used for data collection from the experimental and control groups. The data analysis performed using the Kruskal-Wallis test showed that the experimental group outperformed the control groups in terms of conceptual and procedural knowledge. Furthermore, the results suggest that discourse-based instruction when appropriately designed and implemented can increase students’ understanding of mathematical topics.     

数学理解与理解记忆

数学学习强调理解,理解是数学学习的关键。理解记忆是学习数学的一种很重要的方法。文章就数学理解的作用与理解记忆的重要性作一简要论述。