Making sense of student mathematical thinking: the role of teacher mathematical thinking

Educational Studies in Mathematics - In mathematical whole-class discussions, teachers can build on various student ideas and develop these ideas toward mathematical goals. This requires teachers...     

Algebraic and fractional thinking in collective mathematical reasoning

Educational Studies in Mathematics - This study examines the collective mathematical reasoning when students and teachers in grades 3, 4, and 5 explore fractions derived from length comparisons, in...     

Cultivation of students' mathematical thinking

思维能力对学生智力的发展非常重要,发展和培养学生的思维能力是改革当前小学数学教学的一个重要课题.几年来,我在自己的数学教学工作中,对如何启发学生思考,发展学生的思维,做了不少工作,有一些体会.     

Professional noticing of coordinated mathematical thinking

Over the past three decades, research and policy in many geographic regions has promoted a shift from direct, lecture-oriented mathematics instruction to inquiry-based, dialogic forms of instruction. While theory and research support dialogic instructional approaches, some have noted that the complexities of dialogic teaching make it difficult for teachers to implement. One mechanism by which teachers can improve their decision-making practices in dialogic classrooms is learning to notice (i.e. becoming aware of learners’ processes). While research has contributed frameworks for understanding how teachers notice individual learners’ mathematical thinking, there is little conceptualization regarding how teachers notice group processes in mathematics classrooms, which is integral to dialogic instruction. We offer a noticing framework termed professional noticing of coordinated mathematical thinking that describes how teachers notice group activity in mathematics classrooms. Professional noticing of coordinated mathematical thinking is conceptualized as a bi-dimensional process: noticing groups’ mathematical activity and noticing groups’ coordinated activity. Teachers must become aware of how groups approach the mathematical and collaborative nature of a task, since both of these aspects inform whether learners develop opportunities to learn in groups. The framework describes noticing practices integral to dialogic instruction and promotes inquiry for future research related to teaching moves in dialogic classrooms.     

试论数学教学中数学思维的培养

在进行高等数学的具体授课过程中切实转变教育观念,将培养学生的创新精神和探索精神作为重点来抓,是实现学生数学素养全面提高的重要方法。高等数学作为现今大学科目设置的一门基础课程,授课教师在进行教学活动理应通过有意识的引导,培养学生尽快形成并熟练运用逆向思维能力、类比思维能力、归纳思维能力进行高等数学学习,为学生实现数学素质的全面提高奠定坚实基础。     

Emergent mathematical thinking in the context of play

In the attempt to improve mathematical thinking for safeguarding our future societal needs, there is a worldwide tendency in schools to start training mathematical and arithmetical operations at an earlier age in children’s development. Recent theoretical developments and empirical research have pointed to alternative ways of approaching early mathematical thinking. In these latter approaches, mathematical development in the early developmental stages is seen as an emerging process in the context of children’s own activities that contributes to meaningful learning and stimulation of children’s cultural identity (Bildung approach). The discussion between the training approach versus the ‘Bildung’ approach is still intemperately going on. In this article, some outcomes of a research programme (based at the Free University Amsterdam) are discussed that present empirical studies and their theoretical background (cultural–historical theory, elaborated in an educational concept called ‘Developmental Education’) that demonstrates the promising potentials of promoting mathematical thinking through supporting young children’s appropriation of schematic representations and notations in the context of play.     

论数学教学中学生思维品质的培养

综合教育理论与教学实践 ,论述了在数学教学中培养学生思维品质的新思想和新方法 ,提出了培养学生思维品质的具体措施。     

Process-oriented instruction in learning and thinking strategies

Process-oriented instruction is defined as instruction aimed at teaching thinking strategies and domain-specific knowledge in coherence. This new conception of instruction is derived from psychological research on the way students learn and on the interplay between self-regulation and external regulation of learning. In the research reported here the learning effects of a process-oriented instructional program for university students were empirically studied. The instructional program consisted of a diagnostic learning style instrument, a learning guide and tutorials. The results showed that the majority of students reported significant general, knowledge, insight and application learning effects. The program effects were typified more by integrating and making usable metacognitive knowledge already present, than by increasing knowledge about new subjects. Evidence for transfer effects was obtained because participants in the program scored better than non-participants on two exams of another course. The learning effects were higher than the effects of an preliminary version of the program implemented with students from an open university. These results support the importance of the process-oriented instructional model. The linking of a thorough diagnosis of personal learning styles to individually tailored instructional measures, turned out to be a powerful way to activate students to reflect on their learning and to develop their mental models of learning.     

用数学思想方法探求函数最值

最值问题是在生产、科学研究和日常生活中常会遇到的一类特殊的数学问题 ,尽管其严格的理论指导需借助高等数学知识 ,但由于它与中学数学中许多的知识以及蕴含在这些知识中的数学思想方法紧密相关 ,训练思维能力效果显著 ,所以在高考和数学竞赛中占有相当重要的地位 ,成为近几年高考的热点内容之一。因此在高考数学总复习中 ,对解决最值问题的常用方法进行系统总结 ,并进行深化训练 ,从而提高学生解决综合问题的能力很有必要。本文就利用数学思想方法探求函数最值的常用方法进行归纳整理并举例说明之。一、利用巡数与方程的思想函数的思想 ,…     

论数学思维能力中的分类思想

数学思想是数学知识和方法在更高层次上的抽象与概括,分类是一种重要的数学思想,分类思想是根据数学本质属性的相同点和不同点,将数学研究对象分为不同种类的一种数学思想。分类以比较为基础,比较是分类的前提,分类是比较的结果。分类应在同一标准下具有完备性和互斥性,不同的类之间的逻辑关系是＂或＂,因此,分类的结果是集合运算的＂并＂,通过分类思想的学习,培养学生思维的条理性,缜密性,提高学生的思维能力。初中数学中,分类问题总体归结为两类,涉及数与代数、空间与图形,因此,分类思想是安徽省中考每年必考的核心思想方法。     

英语研究性学习中的批判性思维

研究性学习要求学习主体充分发挥观察力、注意力、记忆力、思维力和想象力来获取信息和知识,其中批判性思维起到导向作用,可见,研究性学习与批判性思维的培养密切相关,而且两者结合得十分自然和谐.研究性学习离不开批判性思维,离开了批判,就无从开展研究.批判性思维的培养,尤其就外语专业学生而言,更离不开研究性学习,在研究中学语言,在学语言中发现问题,在研究问题中培养思辨能力,从而使外语专业的学生成为具备高级思辨能力的有用人才.     

Uses of video in understanding and improving mathematical thinking and teaching

This article characterizes my use of video as a tool for research, design and development. I argue that videos, while a potentially overwhelming source of data, provide the kind of large bandwidth that enables one to capture phenomena that one might otherwise miss; and that although the act of taping is in itself an act of selection, there is typically enough shown in a video that it rewards multiple watching and supports the kinds of arguments over data that are essential for theory testing and replication. In pragmatic terms, video presents phenomena in ways that have an immediacy that is tremendously valuable. I discuss ways in which videos help students and teachers focus on phenomena that might otherwise be very hard to grapple with. This article begins with a brief review of my uses of video, almost 40 years ago, for research and development in problem solving. It then moves to the discussion of very fine-grained research on learning and decision making. The bulk of the article is devoted to a discussion of the teaching for robust understanding (TRU) framework, which was derived in large measure from the extensive review of classroom videotapes, and which serves as the basis for an extensive program of pre-service and in-service professional development. The professional development relies heavily on the use of videos to convey the key ideas in TRU, and to help teachers plan and review instruction.     

数学学习过程中的探索性思维研究

探索性思维,是指"对未知问题或规律寻求认识和解决的思维活动,是一个多环节,多层次的思维体系"。数学的学习过程本身就是不断的进行探索性思维的过程,通过数学的学习能很好的培养学生的探索性思维能力。     

试析《数学分析》的数学思想方法特点

本较详细地分析了《数学分析》课程中的数学思想方法特点，指出重点重数学方法的教学，有助于学生形成良好的数学认知结构,有利于发展学生的思维能力。     

研究性学习实践与思考

研究性学习对于学生的学习方式有深远的意义,已作为一门课程在普通高中开设了几年时间。就目前的情况来看,无论是高中学生还是授课教师普遍缺乏有关课题研究的经验与方法,在实际操作中往往会出现各种问题,因此应该加强指导。     

数学教学中创新思维能力的培养

发展智力、培养创新思维能力是中学数学教师的主要任务，结合教学实践，从4个方面论述了数学教学中如何培养学生的创新思维和创新能力．     

例谈化学解题中的数学思想

近年来,随着高考注重“基础、能力、思想”的考查,在中学化学教学中也愈来愈重视对学生“用数学”的意识和能力的培养。“将化学问题抽象为数学问题,利用数学工具,通过计算和推理（结合化学知识）解决化学问题的能力”成为中学化学思维能力培养的重要方面。因此,熟悉数学思想,有意识地运用数学方法灵活解决有关化学问题,将有利于提高解题的思维能力和技能、技巧。本文通过几例来浅谈一下在化学解题中常见的数学思想,以资学习者参考。