一道题衍生一类题——对一道教材习题的深度开发与探索

数学教材是组织数学教学活动的基本依据,教材习题是数学学习活动的重要载体.本文对一道教材习题进行深度开发与探索,挖掘其潜在知识,抽象其数学本质,提炼其数学精髓,探索其教育价值,实施创造性的教学.     

加快迭代收敛速度的两个命题

定义了单调收敛函数和交错收敛函数,并根据其收敛特点,提出并证明了加快其收敛速度的两个命题.算例表明其效果较好.     

简议焦点三角形

所谓焦点三角形,系指有心圆锥曲线（椭圆、双曲线）上任一点与其两焦点连接构成的三角形．因为焦点三角形是具有特殊意义的三角形,所以它既具有一般三角形的性质,又有其特殊性质．因而解决焦点三角形问题,要紧紧抓住其本质特征（顶点为两焦点和圆锥曲线上的点）,挖掘其内涵、张扬其外延;     

给一组高考题“卸妆”

审视很多高考试题,往往是对某些简单的问题的有意“妆饰”,一旦对其卸了“妆”,便还原了问题的本来面目,使其水落而石出,不难求解．本文对一组高考题卸妆还原,重现其原貌．     

一类具独立子系统的退化时滞控制系统的能控性

讨论退化时滞微分控制系统的能控性问题.首先将退化时滞微分控制系统化为标准形式,除去关联项,得到具独立子系统的退化时滞微分控制系统.然后就一般的退化时滞微分控制系统,得到其能控的充要条件为其可达集等于全空间.对于具独立子系统的广义时滞控制系统,给出其能控的充要条件为每个子系统的可达集等于其相应的子空间,并给出其能控的代数判据,最后举例说明主要结果的应用.     

构造辅助函数计算行列式

对于某类特殊行列式,根据其自身特点,构造相应辅助函数,借以计算其结果。     

Lüroth误差和函数的若干性质

本文研究了Lüroth展式的误差和函数.利用误差和函数的Perron-Frobenius算子,得到其积分值.最后,考察并获得其介值性定理,从而得出其图形是一个分形.     

二次紧支撑样条小波插值及其应用

以二次紧支撑样条小波为基函数,构造了一类二次紧支撑样条小波插值函数,仔细讨论了其计算过程和误差.再将其应用于数值积分,给出了一类求数值积分的新公式,分析了其误差,最后给出一个数值例子.     

Lǖroth误差和函数的若干性质

本文研究了L櫣roth展式的误差和函数.利用误差和函数的PerronFrobenius算子,得到其积分值.最后,考察并获得其介值性定理,从而得出其图形是一个分形.     

Lüroth误差和函数的若干性质

本文研究了Lüroth展式的误差和函数.利用误差和函数的Perron-Frobenius算子,得到其积分值.最后,考察并获得其介值性定理,从而得出其图形是一个分形.     

数学学习与数学思想方法

数学学习,实际上就是对数学知识的理解和对数学思想方法的掌握与运用,数学思想方法是对数学知识的概括,也是数学知识的本质所在.对数学思想方法进行层次性划分,使数学学习具有针对性,同时也从方法论角度提供了数学学习的方法.全面地掌握数学方法,不仅有助于对数学知识的理解和运用,能有效提高数学的学习效率,对提高个体的整体素养也具有重要的现实意义.     

Teachers’ mathematical activity in inquiry-oriented instruction

This work investigates the relationship between teachers’ mathematical activity and the mathematical activity of their students. By analyzing the classroom video data of mathematicians implementing an inquiry-oriented abstract algebra curriculum I was able to identify a variety of ways in which teachers engaged in mathematical activity in response to the mathematical activity of their students. Further, my analysis considered the interactions between teachers’ mathematical activity and the mathematical activity of their students. This analysis suggests that teachers’ mathematical activity can play a significant role in supporting students’ mathematical development, in that it has the potential to both support students’ mathematical activity and influence the mathematical discourse of the classroom community.     

数学分析教学与三种基本数学能力的培养

基本的专业数学能力可分为三个方面:数学发现能力,数学论证能力和数学表达能力.本文结合数学分析课程的教学实践,阐述通过具体教学环节,贯彻培养三种能力的有效途径和方法.     

Connecting mathematical creativity to mathematical ability

This study aims to investigate whether there is a relationship between mathematical ability and mathematical creativity, and to examine the structure of this relationship. Furthermore, in order to validate the relationship between the two constructs, we will trace groups of students that differ across mathematical ability and investigate the relationships amongst these students’ performance on a mathematical ability test and the components of mathematical creativity. Data were collected by administering two tests, a mathematical ability and a mathematical creativity test, to 359 elementary school students. Mathematical ability was considered as a multidimensional construct, including quantitative ability (number sense and pre-algebraic reasoning), causal ability (examination of cause–effect relations), spatial ability (paper folding, perspective and spatial rotation abilities), qualitative ability (processing of similarity and difference relations) and inductive/deductive ability. Mathematical creativity was defined as a domain-specific characteristic, enabling individuals to be characterized by fluency, flexibility and originality in the domain of mathematics. The data analysis revealed that there is a positive correlation between mathematical creativity and mathematical ability. Moreover, confirmatory factor analysis suggested that mathematical creativity is a subcomponent of mathematical ability. Further, latent class analysis showed that three different categories of students can be identified varying in mathematical ability. These groups of students varying in mathematical ability also reflected three categories of students varying in mathematical creativity.     

Teachers creating mathematical models to fairly distribute school funding

Our study aims to investigate what teachers do as they draw on their mathematical understanding and personal experiences to engage in social justice-oriented mathematical modeling. We analyze what ideas were expressed by teachers regarding their mathematical identities while they explore, wrestle with, and reconcile the underlying societal values that support mathematical models. We invited groups of teachers to make mathematical models for distributing school funding given real data from diverse, anonymized schools. Our results show that teachers created and refined diverse mathematical models to connect the mathematical world and societal space and these models reflected different societal values. Drawing on their own experiences, teachers expressed a sense of agency and critical consciousness while making decisions about school funding. This study delineates mathematical contents and processes necessary for advancing a societal goal of fairly distributing funds and we explore how teachers connect to this context as learners and members of society.     

Analyzing students’ communication and representation of mathematical fluency during group tasks

Findings discussed in this paper are from a larger research project exploring mathematical fluency characteristics, and teacher noticing and interpreting of mathematical fluency. The current study involved students from seven primary classes (Kindergarten – Grade 6, N = 63 students) and investigated students’ written work samples and oral discussions as they collaborated in small groups to solve mathematical tasks. Students displayed mathematical fluency both orally and in written/drawn form. Certain aspects of mathematical fluency were easier to identify orally (adaptive reasoning) particularly for younger students and when students did not provide any written reasoning. Analyzing the oral responses was often needed to identify mathematical fluency beyond knowledge of a correct procedure (strategic competence). Findings suggested that the various representations students used were valuable for observing mathematical fluency. These results suggest that oral assessments as a means to understand and interpret students’ mathematical fluency are necessary.     

地方高校数学建模教学模式的探索与实践

针对地方高校学生的特点和培养目标,探索了地方高校以数学建模竞赛、数学模型课、数学实验课为中心的数学建模教学模式的步骤、措施和方法;总结了地方高校数学建模竞赛活动的实践经验.     

The mathematical beliefs and behavior of high school students: Insights from a longitudinal study

There is a documented need for more research on the mathematical beliefs of students below college. In particular, there is a need for more studies on how the mathematical beliefs of these students impact their mathematical behavior in challenging mathematical tasks. This study examines the beliefs on mathematical learning of five high school students and the students’ mathematical behavior in a challenging probability task. The students were participants in an after-school, classroom-based, longitudinal study on students’ development of mathematical ideas funded by the United States National Science Foundation. The results show that particular educational experiences can alter results from previous studies on the mathematical beliefs and behavior of students below college, some of which have been used to justify non-reform pedagogical approaches in mathematics classrooms. Implications for classroom practice and ideas for future research are discussed.     

Understanding secondary mathematics: analysis of linguistic difficulties vis‐a‐vis errors

Misconceptions caused by misunderstanding mathematical language are of different types, e.g. interference of mathematical and non‐mathematical meanings, complexity or unfamiliarity of words, improper use of symbols, syntactical misunderstandings and redundancy or inadequacy of data. Fifteen mathematical words were presented to 84 students of 8th grade and the responses of students were analysed. Mathematical items of a Science Talent Search Test were also analysed from the answers of 100 candidates of 8th grade. The analysis casts light on the processes of the development of mathematical concepts which the students learn through the vicissitudes of the interaction of mathematical and non‐mathematical meanings of words which may be familiar or unfamiliar, relevant or irrelevant, and/or distinct or difficult, to the learners. Complications in understanding mathematical concepts for individual students are pointed out from these experiments. Acquaintance with such communication processes of learners can also help in detecting strategies of imparting mathematical instructions to the learners. Roles of the uses of common, mathematical, transformational, and story‐telling language have also been discussed. The paper concludes with some comments on the importance of guided discovery learning, error analysis by teachers, and the preparation of a register of mathematical language.