Recognising mathematical creativity in schoolchildren

Examples of tasks designed to recognise creative thinking within mathematics, used with 11–12-year-old pupuls, are described. The first construct empoyed in the design of these tasks is the ability to overcome fixation. Sometimes pupils demonstrate content-universe fixation, by restricting their thinking about a problem to an insufficient or inappropriate range of elements. Other times they show algorithmic fixation by continuing to adhere to a routine procedure or stereotype response even when this becomes inefficient or inappropriate. The second construct employed is that of divergent production, indicated by flexibility and originality in mathematical tasks to which a large number of appropriate responses are possible. Examples of three categories of such tasks are described: (1) problem-solving, (2) problem-posing, and (3) redefinition. Examples of pupils’ responses to various tasks are used to argue that they do indeed reveal thinking that can justifiably be described as creative. The relationship to conventional mathematics attainment is discussed-mathematics attainment is seen to limit but not to determine mathematical creativity.     

Students' Accuracy of Measurement Estimation: Context,Units, and Logical Thinking

This study examined students' accuracy of measurement estimation for linear distances, different units of measure, task context, and the relationship between accuracy estimation and logical thinking. Middle school students completed a series of tasks that included estimating the length of various objects in different contexts and completed a test of logical thinking ability. Results found that the students were not able to give accurate estimations for the lengths of familiar objects. Students were also less accurate in estimating in metric units as compared to English or novel units. Estimation accuracy was dependent on the task context. There were significant differences in estimation accuracy for two‐ versus three‐dimensional estimation tasks. There were no significant differences for estimating objects with different orientations or embedded objects. For the tasks requiring the students to estimate in English units, the embedded task and the three‐dimensional tasks were correlated with logical thinking. For estimation tasks with novel units, three‐dimensional and two‐dimensional estimation tasks were significantly correlated with the logical thinking. In order to interact effectively with our environment it is essential to possess an intuitive grasp of both dimension and scale and to be able to manipulate such information. Estimation, approximating and measuring are all components of such intuition ( Forrester, Latham, & Shire, 1990 , p. 283).     

供需模型在众包平台定价中的应用

以众包平台为对象,基于已有数据,考虑供求关系,通过因子分析法对原始数据加以处理,建立任务的定价模型.在实际应用中,可以对分布密集的任务进行打包,并给出针对这种情况的定价模型.结果表明:提出的模型不仅考虑了会员人数与任务数量的关系,还考虑了个人需求的差异,提高了任务的完成度,对众包平台定价的发展与完善有着重要的意义.     

微积分全英语教学模式探讨——以电子科技大学格拉斯哥学院课程Calculus(微积分)为例

电子科技大学格拉斯哥学院电子信息工程专业是电子科技大学与英国格拉斯哥大学的联合办学项目,Calculus(微积分)课程是大一学生的一门重要的基础课程,该课程采用全英语教学.全英语教学是目前我国高水平办学的教学模式之一,在教材选择、课程大纲、授课模式上都有其独特性.为了更有效地提高该课程的教学质量和教学效果,本文对微积分全英语教学的现状、价值意义、存在问题、教学模式等方面进行探讨,并对存在的主要问题给出了一些具体的方案.     

Mathematics education and democracy

In this paper, we investigate the relationship between mathematics education and the notions of education for all/democracy. In order to proceed with our analysis, we present Marx’s concept of commodity and Jean Baudrillard’s concept of sign value as a theoretical reference in the discussion of how knowledge has become a universal need in today’s society and ideology. After, we engage in showing mathematics education’s historical and epistemological grip to this ideology. We claim that mathematics education appears in the time period that English becomes an international language and the notion of international seems to be a key constructor in the constitution of that ideology. Here, we draw from Derrida’s famous saying that “there is nothing beyond the text”. We conclude that a critique to modern society and education has been developed from an idealistic concept of democracy.     

The Japanese perspective on TIMSS

The International Association for the Evaluation of Educational Achievement (IEA) undertook three international mathematics studies: the first, FIMS, in 1976, the second, SIMS, in 1980, and the third, TIMSS, in 1995, 13-year-old Japanese students were included in these studies. The purpose of this paper is to discuss the level of mathematics achievement and attitudes toward mathematics among Japanese students and the situation of mathematics education in Japan, based on the results of TIMSS and previous studies. From our analysis of results, we can indicate the following points: although achievement in the fundamental techniques of calculation can be viewed in general as satisfactory, the attainment levels cannot be regarded as acceptable for problems which require a high degree of thinking and comprehension. And from the consequence of international comparison about interest and attitude, it became evident that a smaller fraction of Japanese students have a favorable opinion than in other countries.     

Combining the assumptions of variable and constant returns to scale in the efficiency evaluation of secondary schools

Our paper reports on the use of data envelopment analysis (DEA) for the assessment of performance of secondary schools in Malaysia during the implementation of the policy of teaching and learning mathematics and science subjects in the English language (PPSMI). The novelty of our application is that it makes use of the hybrid returns-to-scale (HRS) DEA model. This combines the assumption of constant returns to scale with respect to quantity inputs and outputs (teaching provision and students) and variable returns to scale (VRS) with respect to quality factors (attainment levels on entry and exit) and socio-economic status of student families. We argue that the HRS model is a better-informed model than the conventional VRS model in the described application. Because the HRS technology is larger than the VRS technology, the new model provides a tangibly better discrimination on efficiency than could be obtained by the VRS model. To assess the productivity change of secondary schools over the years surrounding the introduction of the PPSMI policy, we adapt the Malmquist productivity index and its decomposition to the case of HRS model.     

航空公司中的飞行员配置研究

随着新航线的开辟和新飞机的投入使用,航空公司的任务量急剧增长,飞行员需求的估计和机组的合理配置变得日益重要。本文基于实际工作需求,提出一类用于评估飞行员数量,并进行机组均衡配置的整数规划模型,为航空公司进行飞行员的数量评估和任务分配提供决策支持。本文在以下两个方面做出改进:1)加入国际长途航线,研究国内短途航线和国际长途航线混合搭配情况下飞行员的配置问题;2)鉴于大型整数规划求解的复杂性,通过评估航空公司所需要的飞行员数量的下界,为决策人员制定中长期规划提供依据。本文通过启发式算法进行飞行员的配置,计算结果接近理论问题的下界,证明了算法的有效性。     

Elementary teachers’ mathematical beliefs and mathematics anxiety: How do they shape instructional practices?

This quantitative study investigated the relationships among practicing elementary teachers’ (N = 153) beliefs about mathematics and its teaching and learning, mathematics anxiety, and instructional practices in mathematics. When viewed singly, the findings reveal the teachers with higher levels of mathematics anxiety tend to use less standards‐based instruction and those with beliefs oriented toward a problem‐solving view of mathematics reported more standards‐based teaching. A combined analysis shows that after controlling for mathematical beliefs, teaching longevity, and educational degree attainment, there is no relationship between teachers’ mathematics anxiety and instructional practices. These findings suggest a spurious relationship between anxiety and practices, with beliefs having the strongest relationship with practices. Several suggestions for positively influencing the mathematical beliefs and affect in general of elementary teachers while learning mathematics are offered.     

On the eigenproblem for normal matrices

Summary The Householder-Givens method for the solution of the Hermitean eigenproblem is used for the solution of the skew-Hermitean and the normal eigenproblem.This research was performed at the (Mathematical Institute of the) State University in Utrecht.     

Cognitive styles,task presentation mode and mathematical performance

This paper reports the outcomes of an empirical study undertaken to investigate the relationship of prospective teachers’ cognitive styles and levels of performance in measurement and spatial tasks. A total of 116 prospective kindergarten school teachers were tested using the VICS and the extended CSA-WA tests (Peterson 2005) in order to place them along the Verbal/Imagery and the Wholistic/Analytic cognitive style continua. The same prospective teachers were also administered a mathematical test with 6 measurement and 6 spatial tasks. The results suggest that there were no significant differences between Verbalisers-Imagers and Wholistic-Analytic prospective teachers in their performance on the spatial pictorial and textual tasks, and on the measurement textual tasks. However, there were differences between Verbalisers-Imagers and Wholistic-Analytic prospective teachers in their performance on the measurement pictorial tasks. This difference was attributed to the performance of low achievers. High achievers performed in the same way independently of their cognitive styles.     

Spatial visualizers, object visualizers and verbalizers: their mathematical creative abilities

This paper investigates the relationship between the creative process in mathematical tasks and spatial, object and verbal cognitive styles. A group of 96 prospective primary school teachers completed the Object-Spatial Imagery and Verbal Questionnaire and took a mathematical creativity test. The results of a multiple regression analysis demonstrated that whereas visual cognitive styles (spatial and object imagery) were statistically significant predictors of participants’ creative abilities in mathematics, verbal cognitive style did not predict these abilities. Further analysis of the data indicated that spatial imagery cognitive style was related to mathematical fluency, flexibility and originality. On the other hand, object imagery cognitive style was negatively related to mathematical originality and verbal cognitive style was negatively related to mathematical flexibility. The study also revealed that individuals with a tendency towards different cognitive styles employed different strategies in the creative mathematical tasks.     

International publishing in operational research

The primary literature in OR, as defined in International Abstracts in Operations Research, has grown by 47 per cent between 1975 and 1985. EURO and North-America together account for 85 percent of the primary literature. The USA is more than three times as productive per capita as EURO. English has become the standard language; its share in the primary literature increased from 86 to 93 per cent. Most primary journals are nationalist and many cannot be called international. These and more detailed results are presented in this cross-section analysis of the OR primary literature.     

Lambda pruning: an approximation of the string subsequence kernel for practical SVM classification and redundancy clustering

The support vector machine (SVM) is a powerful learning algorithm, e.g., for classification and clustering tasks, that works even for complex data structures such as strings, trees, lists and general graphs. It is based on the usage of a kernel function for measuring scalar products between data units. For analyzing string data Lodhi et al. (J Mach Learn Res 2:419–444, 2002) have introduced a String Subsequence kernel (SSK). In this paper we propose an approximation to SSK based on dropping higher orders terms (i.e., subsequences which are spread out more than a certain threshold) that reduces the computational burden of SSK. As we are also concerned with practical application of complex kernels with high computational complexity and memory consumption, we provide an empirical model to predict runtime and memory of the approximation as well as the original SSK, based on easily measurable properties of input data. We provide extensive results on the properties of the proposed approximation, SSK-LP, with respect to prediction accuracy, runtime and memory consumption. Using some real-life datasets of text mining tasks, we show that models based on SSK and SSK-LP perform similarly for a set of real-life learning tasks, and that the empirical runtime model is also useful in roughly determining total learning time for a SVM using either kernel.     

Mathematical creativity in generally gifted and mathematically excelling adolescents: what makes the difference?

Due to uncertainty regarding the relationship between mathematical creativity, mathematical expertise and general giftedness, we have conducted a large-scale study that explores the relationship between mathematical creativity and mathematical ability. We distinguish between relative and absolute creativity in order to address personal creativity as a characteristic that can be developed in schoolchildren. This paper presents part of a study that focuses on the power of multiple solution tasks (MSTs) as a tool for the evaluation of relative creativity. We discuss relationships between mathematical creativity, mathematical ability and general giftedness as reflected in the present empirical study of senior high school students in Israel which implemented the MST tool. The study demonstrates that between-group differences are task dependent and are a function of mathematical insight as it is integrated in the mathematical task. Thus, we conclude that different types of MSTs can be used for different research purposes, which we discuss at the end of this paper.     

基于广义脉冲的东北三省外贸外资与经济增长研究

作为连接其他国家和地区的桥梁和纽带,对外经贸在国际经济关系中有着十分重要的地位。本研究选取对外经贸与经济增长代表性指标,以各指标平均增长率为切入点,通过构建相对增长指数模型,测算东北三省的外贸相对增长指数(TGI)和外资相对增长指数(CGI)并进行分类,结果显示三省外贸、外资增长类型截然不同。进一步,将历年外贸、外资增速与经济增速进行对比分析发现,外贸增速与经济增速具高度同步性,且始终扮演经济晴雨表的角色,外资增速则更具不确定性和偶然性。最后,广义脉冲响应分析显示外贸外资冲击经济增长均产生显著效应,但响应程度不尽相同,各指标贡献度通过方差分解呈现。本文为外贸外资与经济增长关系研究提供了新的视角。     

THE IMPACT OF THE NATIONAL NUMERACY STRATEGY IN YEAR 4 (I): ATTAINMENTAND LEARNING

The Nuffield Year 4 project set out to examine selected aspects of the impact of the National Numeracy Strategy (NNS) in primary schools by using comparable data collected from 35 schools in 1997/98 and 2001/02 as part of the Leverhulme Numeracy Research Programme. In this first paper we report on changes in attainment using a specially designed test of numeracy administered in October and June. While numeracy attainment rose during the period of the introduction of the NNS, the effect was small and it varied in relation to different aspects of the curriculum. The results of low attainers fell on average, as did results in over a third of schools. The amount of progress made over Year 4 was similar in 2001/02 and in 1997/98. Reasons for these changes are explored.     

Community detection-based deep neural network architectures: A fully automated framework based on Likert-scale data

Deep neural networks (DNNs) have emerged as a state-of-the-art tool in very different research fields due to its adaptive power to the decision space since they do not presuppose any linear relationship between data. Some of the main disadvantages of these trending models are that the choice of the network underlying architecture profoundly influences the performance of the model and that the architecture design requires prior knowledge of the field of study. The use of questionnaires is hugely extended in social/behavioral sciences. The main contribution of this work is to automate the process of a DNN architecture design by using an agglomerative hierarchical algorithm that mimics the conceptual structure of such surveys. Although the train had regression purposes, it is easily convertible to deal with classification tasks. Our proposed methodology will be tested with a database containing socio-demographic data and the responses to five psychometric Likert scales related to the prediction of happiness. These scales have been already used to design a DNN architecture based on the subdimension of the scales. We show that our new network configurations outperform the previous existing DNN architectures.