Supervision in Secondary Education

It is widely accepted that including writing activities in the learning process positively impacts student achievement and leads to greater depth of student understanding. This writing is often missing in the math classroom though, when the focus is misplaced on rote procedures. In these classrooms students learn mathematical processes but have little depth of understanding into the mathematical foundations, nor have an ability to clearly express their mathematical reasoning. This article promotes the use of Internet-based chat, forums, and blogs as the environment in which necessary mathematical writing can occur. Zemelman, Daniels, and Hyde provide a best practice framework through which the benefits of chat, forum, and blog writing are obvious. Student engagement with material increases in a cooperative environment, where a real audience and purpose for writing is clear, and student ownership in personal learning grows. In addition, students mature in traditional reading and writing literacy and further develop critical thinking skills.     

The Role of Proof in Comprehending and Teaching Elementary Linear Algebra

We describe how elementary Linear Algebra can be taught successfully while introducing students to the concept and practice of ‘mathematical proof’.This is done badly with a sophisticated Definition–Lemma–Proof–Theorem–Proof–Corollary(DLPTPC) approach; badly – since students in elementary Linear Algebra courses have very little experience with proofs and mathematical rigor. Instead, the subjects and concepts of Linear Algebra can be introduced in an exploratory and fundamentally reasoned way. One seemingly successful way to do this is to explore the concept of solvability of linear systems first via the row echelon form (REF). Solvability questions lead to row and column criteria for a REF that can be used repeatedly to: compute subspaces, settle linear (in)dependence, find inverses, perform basis change, compute determinants, analyze eigensystems etc. If these subjects are explained heuristically from the first principles of linear transformations, linear equations, and the REF, students experience the power of a concept–built approach and reap the benefit of deep math understanding. Moreover, early ‘salient point’ proofs lead to an intuitive understanding of ‘math proof’. Once the basic concept of ‘proof’ is ingrained in students, more abstract proofs, even DLPTPC style expositions, on normal matrices, the SVD etc. become accessible and understandable to sophomore students. With the help of this gentle early approach, the concept and construct of a ‘math proof’ becomes firmly embedded in the students' minds and helps with future math courses and general scientific reasoning. This revised version was published online in July 2006 with corrections to the Cover Date.     

中学生数学学习中抽象概括的思维障碍研究

数学活动的抽象概括过程分为抽离、筛选,假设、扩张和确认4个阶段.学生进行数学活动中抽象概括的思维障碍有:抽离的意识,抽离的方向,抽离的表达;筛选的意识,筛选的逻辑方向,筛选的逻辑基础及对应的语言表达;扩张的意识,扩张的表达形式;确认的意识,确认的逻辑,确认的形式具体化.初中学生数学学习中抽象概括的思维障碍相当严重,教师在教学中应对此有足够的估计.     

Working memory and mathematics: A review of developmental,individual difference,and cognitive approaches

Working memory refers to a mental workspace, involved in controlling, regulating, and actively maintaining relevant information to accomplish complex cognitive tasks (e.g. mathematical processing). Despite the potential relevance of a relation between working memory and math for understanding developmental and individual differences in mathematical skills, the nature of this relationship is not well-understood. This paper reviews four approaches that address the relation of working memory and math: 1) dual task studies establishing the role of working memory during on-line math performance; 2) individual difference studies examining working memory in children with math difficulties; 3) studies of working memory as a predictor of mathematical outcomes; and 4) longitudinal studies of working memory and math. The goal of this review is to evaluate current information on the nature of the relationship between working memory and math provided by these four approaches, and to present some of the outstanding questions for future research.     

数学思想在初中教学中的渗透

这是一项对“数学思想与初中学生巧学”的研究。数学思想方法是数学思维的内核,它们是隐含于概念的形成,公式的论证,习题的解决之中的,所以数学教师要把散见于习题中的数学思想整理出来提供学生学习。笔者主要联系数形结合、类比、分类讨论、函数、化归等思想,结合本人教学经验,讲述如何在教学中有效地渗透数学思想。加深学生对数学思想的理解,提高解题技能,完善数学思维。     

基于问题解决的数学理解性教学设计

数学理解性教学设计是将数学理解性教学由目前的理想状态转化为教学现实的最佳途径。基于问题解决的数学理解性教学设计包含分析、评价、设计、开发和实施五个基本要素,这些要素构成相互关联的循环系统。在整个系统设计中应以学生为中心、以问题为引导、依据数学理解性教学所基于的学与教的原理进行设计。     

Exploring student beliefs and understanding in elementary science and mathematics

This study had the goal of investigating the association among elementary students' (N = 276) science and math beliefs and the relationship between those beliefs and teachers' ratings of mathematical and science understanding. Results of structural path analysis indicate that in science, intellectual risk‐taking (IRT; the willingness to share tentative ideas, ask questions, attempting to do, and learn new things) was positively related to teachers' ratings of science understanding, while creative self‐efficacy (CSE) beliefs (i.e., students' confidence in their ability to generate ideas and solutions in science) were indirectly related (working through IRT). Results also indicate that students' scientific certainty beliefs (i.e., the belief that science knowledge is stable, fixed, and represented by correct answers) were negatively related to teachers' ratings of science understanding. With respect to math, results indicate that students' CSE beliefs were positively related to teachers' ratings of math understanding; whereas students' mathematical source beliefs (i.e., believing that math knowledge originates from external sources) were negatively related. © 2012 Wiley Periodicals, Inc. J Res Sci Teach 49: 942–960, 2012     

高职学生导数应用能力的培养

应用导数知识解决实际问题,是高职学生学习高等数学,形成数学应用能力的重要方面.本文通过一些实例展示导数的数学美和导数解决实际问题的活力,以期使高职学生充分了解导数学习的重要性,提高高职学生学习高等数学的自觉性和主动性.     

生态学“抽象化”数学建模实在性分析

在科学建模中,由于目标系统的复杂性,人们往往借助于“抽象化”和“理想化”方法来研究对象。“理想化”是对系统刻意地“扭曲”表征,“抽象化”又遗漏了诸多细节。很多人质疑道,这种简化的数学模型无法正确地反映真实的生态学系统,建模的抽象化和理想化不能保证模型的实在性。这是生态学数学模型的反实在论。“抽象化”包含“遗漏细节”抽象化和“聚集意义”抽象化两种形式。抽象化建模是一种特定的视角来整合所表征生态系统中的相关因素,获得理论上的普遍性,进而能够追求实在。     

剖析一道数学题蕴含的数学思想

数学题的求解过程是一个运用数学思想、方法的过程.而人们多注重解题的过程,忽视了其中蕴含的数学思想,淡化了对数学思想的认识,使数学思想的理论与实际脱节,反映了数学教学的不完善,数学思想是数学的精髓,它对数学问题的解决起着高层次的指导作用,是知识转化为能力的桥梁.数学教学如能恰到好处地溶入一些数学思想,不但会加深各科间的纵横联系,提高数学能力,而且还能激发学生学习的兴趣,调动学习的积极性,更好地开辟数学思维的空间.     

有关数学证明的一次教师问卷调查

Mathematical proofs are not only the focus of every country’s mathematics curriculum reforms, but also the subject of research on mathematics education. This paper is based on a survey of mathematics teachers, the goal of which was to investigate the understanding of mathematical proofs of secondary school math teachers, their levels of mathematical proofs, and their ability to comprehensively teach mathematical proofs. Preliminary results of the survey provide insight into several characteristic aspects of Chinese secondary school teachers’ literacy of mathematical proofs.     

Maternal Math Talk in the Home and Math Skills in Preschool Children

Research Findings: The current study analyzed the relation between the amount of mathematical input that preschool children hear (i.e., math talk) from their mothers in their homes and their early math ability a year later. Forty mother–child dyads recorded their naturalistic exchanges in their homes using an enhanced audio-recording device (the Language ENvironment Analysis System). Results from a sample of naturalistic interactions during mealtimes indicated that all mothers involved their children in a variety of math exchanges, although there were differences in the amount of math input that children received. Moreover, being exposed to more instances of math talk was positively related to children’s early mathematical ability a year after the recordings, even after we controlled for maternal education, self-regulation, and recorded minutes. Practice or Policy: These findings improve the understanding of how mothers use math with their preschool children in naturalistic contexts, providing some insight for parents into how to foster children’s math skills through verbal input in their normal routines. Moreover, these findings inform kindergarten teachers and practitioners about the math input that children receive at home, which may encourage them to adapt their practice by considering the home environment.     

Abstraction and Consolidation

The framework for this paper is a recently developed theory of abstraction in context. The paper reports on data collected from one student working on tasks concerned with absolute value functions. It examines the relationship between mathematical constructions and abstractions. It argues that an abstraction is a consolidated construction that can be used to create new constructions. Newly formed constructions are fragile entities. In the course of consolidating a construction this student creates connections between the new construction and already established mathematical knowledge and develops a language to describe and guide mathematical actions related to the construction. The resulting abstraction is a more resilient form of the construction and the student is able to justify his assertions. The paper also examines issues in designing tasks to consolidate a construction.     

数学抽象的思辨

数学抽象体现了人类的活动，从数学抽象的背景、产生、内容、方法和形式等方面探讨数学抽象活动的过程与结果、形式与实质之间的思辨关系是深化数学教育理论研究的需要。     

对中学生数学认知结构建构途径的研究

学生数学认知结构的建构实质是指对数学概念、定理、公式和命题的理解,以及蕴涵其中的思想、方法的掌握和运用。学生数学学习的过程,实际上就是学生自身数学认知结构的不断搭建和完善的过程,是学生数学思维能力不断拓展和发展的过程。本文试就学生数学认知结构的建构过程中的一些途径和操作方法作一些探讨和总结。     

“数学模型”课程教学体会

数学模型对提高学生的综合数学素质具有重要意义。本文在实践的基础上,论述了数学建模在培养学生的综合应用能力方面的重要作用。以提高教学质量和加强学生应用数学能力培养为目标,通过科学选取教学内容,运用互动式教学方式,加强实践力度,重视数学建模竞赛等方面阐述了笔者在数学建模教学中的体会和认识。     

论数学课程与教学的现实性

数学课程编制和数学教学都要注重学生的数学现实。不同学生具有不同的数学现实。注重学生现实,要求课程教学要注意常识的利用。面向学生现实,需要尊重学生解题思维的个性。注意问题的现实性,要求减少假同构问题。现实应是学生的现实,新课程应真正关注学生的现实。     

高职生数学价值观现状的调查与分析

理清数学价值观的概念与内涵,并采用问卷调查法,对高职生的数学价值观现状进行调查与分析,结果表明：高职生在对数学价值的认识上总体偏低,对于数学的实用价值与德育价值有一定的认识,但认识程度不高,对于数学的审美价值缺乏必要的认识。多数高职生认为学习高等数学不很重要,高等数学的学习价值不大。     

数学应用问题解决认知障碍的成因及教学策略

学生解决数学应用问题认知障碍的成因主要有学生的基础知识不牢固、问题背景的复杂性、认知图式检索失败、没有理解和掌握数学思想方法、解题监控能力差等.克服学生解题认知障碍的教学策略主要有:加强数学"双基"的教学;加强数学建模策略与方法的教学;采用样例教学和变式练习;培养学生反思能力和自我监控能力,克服思维定势的消极影响.