质数模的一元同余方程

讨论了质数模的一元同余方程,研究了根与重根.     

The class number problem

This two part article introduces the reader to the notion of ‘class numbers’. The first part defines class numbers the way they arose in the study of ‘binary quadratic equations’.     

The class number problem

The second part gives an introduction to ‘algebraic number theory’, defines class numbers for finite extensions of the field of rational numbers and proves that in the context of quadratic fields, this definition coincides with the definition of class numbers via binary quadratic forms given in the first part.     

一般线性同余方程组有解的判定条件

讨论了一般的线性同余方程组解的存在性问题，给出了一般线性同余方程组是否有解的判别条件。从而推广了引理2的结论．使求解线性同余方程组问题更为方便有效．     

互素模一般线性同余方程式组中推广的孙子定理

讨论了一般线性同余方程组的求解问题,给出了一般同余方程组在有解情形下解的统一的表达式,得到了几个有益的结果,在理论上作了一种新的尝试,从而推广了孙子定理.     

Manipulatives and number sentences in computer aided arithmetic word problem solving

In this study we investigated the relative contribution of two main components often used in the instruction of arithmetic word problem solving to first-grade children and children with learning problems, external representation with manipulatives and formal mathematical representation with number sentences. Four computer aided treatments were developed along these dimensions. Furthermore, an assessment control group was formed.It was shown that performance improved significantly after treatment in comparison to the performance of the assessment control group. Comparison of the different training conditions showed that a number sentence treatment was the most effective training component. However, this effect was only present with the relatively high competent children and not with the relatively low performing children.     

和为偶数的奇素数对的个数

和为偶数N的奇数对可分为三种情况,第一种是奇合数对(这里把1看做奇合数);第二种是1个是奇合数、1个是奇素数的奇数对;第三种是奇素数对.小于N的奇合数的大约个数可以根据奇合数所含的因数情况来求出,和为N的奇合数对的大约个数也可以根据奇合数对所含的因数情况来求出,小于N的奇合数除两两组成和为N的奇合数对外,其余只能与小于N的奇素数组成和为N的奇数对.求出前两种和为N的奇数对的大约个数,就能求出和为N的奇素数对的大约个数.     

The effects of problem format,number of independent variables,and their interaction on student performance on a control of variables reasoning problem

This study was designed to investigate the effects of problem format and the number of independent variables in the problem on the responses of students to a control of variables reasoning task. The Bending Rods problem was used. A four times two factorial research design with four levels (2-, 3-, 4-, 5-variables) of the number of independent variables and two formats (essay and multiple choice) was set up for the investigation. 548 eighth graders in 24 science classes in a suburban public school district northwest of Chicago participated in the study. Students were administered the Bending Rods problem in groups in their science classes. Each classroom was randomly assigned to a cell in the research design. Initial analysis of the data indicated four separate levels (2-, 3-, 4-, 5-variables) of the independent variable number of independent variables in the problem' were unnecessary. The 2- and 3-variable levels were pooled as were the 4- and 5-variable levels to give two distinct levels of this variable. Analysis of variance indicated that: (1) task format had no effect on subject's scores; but (2) the differences between subjects' mean scores on the 2-, 3-variable essay versions and the 4-, 5-variable essay versions are significantly greater than the mean scores of corresponding multiple choice versions of the task, which exhibit rather uniform scores; and (3) the 2-, 3-variable forms together were significantly less difficult for students than were the 4-, 5-variable forms together. Discussion focuses on interpretations that include: (1) task demands on working memory; (2) Pascual-Leone's M-energy; and (3) the use of testwiseness strategies in solving essay and multiple choice tasks, respectively. Implications for science teachers are set forth in view of these discussion points and their relevance to test construction and classroom assessment of learning objectives in science.     

线性同余方程组解的存在性及其算法

讨论一般的一次同余方程组的解的存在性与性与结构，并给出了求解的新算法，这是名的孙子定理的一个推广。     

抑郁个体心境一致性记忆的实验研究

本研究以抑郁大学生为被试，采用最新的研究方法——加工分离程序的扩展模型，更科学地区分了意识和无意识的独立作用，避免了两者的相互感染．研究结果表明：(1)抑郁个体既存在外显的MCM，也存在内隐的MCM；(2)抑郁个体的认知是以“不愉快”为主题、与不被喜欢和被抛弃相联系的。     

《和为偶数的奇素数对的个数》之续篇

本续篇根据素数定理和有关无穷乘积,再度演化和为偶数的奇素数对的个数的求解公式,得出:和为偶数N的奇素数对的个数大于2N/πln2N,并且举几例比较结果.哥德巴赫猜想应该是和为偶数N的奇素数对的个数为1的一个特例。     

一般线性同余方程组有解的判定条件

讨论了一般的线性同余方程组，给出了一般同余方程组的解的判别条件，从而推广了引理2的结论．使解题更为有效．     

Learning verbs more effectively through meaning congruent action animations

The current study investigates the effectiveness of learning words while displaying meaning congruent animations. We explore whether learning words with animation is sensitive to properties known to influence action understanding. We apply an embodied cognition framework and predictions from a recent theory about language and action (Action-Based Language theory, Glenberg & Gallese, 2012). The current study aims to investigate whether dynamic animations add to word learning (Experiment 1) and what the linguistic relation between the dynamic animation and the word learning is (Experiment 2). Results indicate that meaning congruent animations improved verb learning compared to meaning incongruent animations when measured by a recognition task. When measured by an active recall task, congruent animations led to better learning than static pictures. In both measures, meaning congruent animations support word learning. Experiment 2 replicates and extends this and suggests that highlighting conceptual information related to the dynamic action (such as the goal) improves word learning further. The findings are in line with Action-Based Language theory, which suggests that children are able to make better simulations of an action during learning when supported by meaning congruent animations. Highlighting conceptual information additionally supports this learning process.     

Preach what you teach! Teacher educators and congruent teaching

Teacher educators seem to agree that, to be able to support their student teachers' learning, they themselves should be good models of the kind of teaching they are trying to promote. However, it is clear from the literature that this congruent teaching is not self‐evident in teacher education. In the present article, we describe a small in‐depth study, in which we attempted to establish whether teacher educators begin to teach more congruently when supported, and the factors influencing the occurrence or non‐occurrence of such congruent teaching. To do so, we organised a workshop on the subject. Before and after the workshop, we interviewed the participating teacher educators, using videotapes of their lessons. To discover the possible contribution of the workshop to their congruent teaching, we later compared both interviews. We found that a particularly important aspect of congruent teaching, i.e. the teacher educators' ability to link their own teaching to theory, had improved. Our conclusion is that the acquisition of a language enabling them to talk about congruent teaching helps teacher educators to overcome problems with congruent teaching.     

The secretary problem

In many spheres of activity, decisions must often be made under uncertain conditions. One such problem relates to selecting a candidate from a known number when: (a) candidates arrive in a sequence; (b) the selection process has to decide on a candidate then and there; (c) the process terminates if a candidate is selected; (d) the process continues if the candidate is not selected. The question is: What is the strategy that maximises the probability of selecting the best candidate? How does one use the ‘scores’ of each of the candidates seen so far to decide if the present candidate must be selected or if the process be continued, in the quest for the best candidate? This is the problem ofoptimal stopping, an example of which is discussed here.     

The Sierpinski problem

In this article, we describe briefly a number-theoretic problem first studied by Sierpiński, now known as theSierpiński problem. The problem remains open.