On sums of powers of integers

For non-negative integers k, n, let P _k (n) denote the sum {fx27-1}. We show by two different means that if k ≥ 3 and odd, then n ²(n+1)² iss a factor of the polynomial P _k (n); and if k ≥ 2 and even, then n (n+1) (2n+1) is a factor of the polynomial P _k (n). We also derive a relatively unknown result first obtained by Johann Faulhaber in the 17th century. Shailesh Shirali has been at Rishi Valley School, Andhra Pradesh (Krishnamurti Foundation India) since the 1980’s. He has a deep interest in teaching and writing about mathematics at the high school/post school levels, with particular emphasis on problem solving and on historical aspects of the subject. He has been involved in the Mathematics Olympiad movement at the national and international level for the past two decades. He is the author of several expository books and articles aimed at interested high school students.     

Teaching Physics to In-Service Primary School Teachers in the Context of the History of Science: The Case of Falling Bodies

Our paper presents an in-service primary school teachers’ training program which is based on the idea that the history of science can play a vital role in promoting the learning of physics. This training program has been developed in the context of Comenius 2.1 which is a European Union program. This program that we have developed in the University of Athens is based on socioconstructivist and sociocultural learning principles with the intention of helping teachers to appropriate the basic knowledge on the issue of falling bodies. Moreover, it has the aim to make explicit through the exploitation of authentic historical science events, on the above topic (Aristotle’s, Galileo’s and Newton’s theories on falling bodies) the Nature of Science (NoS), the Nature of Learning (NoL) and the Nature of Teaching (NoT). During the implementation of the program we have used a variety of teaching strategies (e.g. group work, making of posters, making of concept maps, simulations) that utilize historical scientific materials on the issue of falling bodies.

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Unit ball in high dimensions

In this article, we compute the volume V _n of the unit ball in an n-dimensional space. For n = 1, 2, 3, the volumes are respectively 2, π 4π /3, which are the length of interval [−1,1], area of a unit circle and volume of the unit sphere. The numbers V _n ‘appear’ to increase. But in fact this not so. In fact V _n tends to zero as n tends to infinity! K B Athreya is a Visiting Professor at IMI, Mathematics Department, IISc, Bangalore. He is a professor in the Mathematics Department of the Iowa State University, USA. His research interests include mathematical analysis, probability theory and its applications and statistics. His spare time is spent listening to Indian classical music.     

A general version of the Morse-Sard theorem

Let k,m,n be positive integers,and k≥2,α∈(0,1],0＜r＜min {m,n} an integer,d=r (m-r)/(k α),and iff∈ Ck,α(Rm,R″),A =Cr(f)= {x ∈ Rm ,rank(Df(x))≤r},thenf(A)is d-null.Thus the statement posed by Arthur Sard in 1965 can be completely solved when k≥2.     

The Notion of Proof in the Context of Elementary School Mathematics

Despite increased appreciation of the role of proof in students’ mathematical experiences across all grades, little research has focused on the issue of understanding and characterizing the notion of proof at the elementary school level. This paper takes a step toward addressing this limitation, by examining the characteristics of four major features of any given argument – foundation, formulation, representation, and social dimension – so that the argument could count as proof at the elementary school level. My examination is situated in an episode from a third-grade class, which presents a student’s argument that could potentially count as proof. In order to examine the extent to which this argument could count as proof (given its four major elements), I develop and use a theoretical framework that is comprised of two principles for conceptualizing the notion of proof in school mathematics: (1) The intellectual-honesty principle, which states that the notion of proof in school mathematics should be conceptualized so that it is, at once, honest to mathematics as a discipline and honoring of students as mathematical learners; and (2) The continuum principle, which states that there should be continuity in how the notion of proof is conceptualized in different grade levels so that students’ experiences with proof in school have coherence. The two principles offer the basis for certain judgments about whether the particular argument in the episode could count as proof. Also, they support more broadly ideas for a possible conceptualization of the notion of proof in the elementary grades.     

Results on energies for trees with a given diameter having perfect matching

Let Γd2nbe the set of trees with a given diameter d having a perfect matching,where 2n is the number of vertex.For a tree T in Γd2n,let Pd+1be a diameter of T and q = d m,where m is the number of the edges of perfect matching inPd+1.It can be found that the trees with minimal energy in Γd2nfor four cases q = d 2,d 3,d 4,[d2],and two remarks aregiven about the trees with minimal energy in Γd2nfor2d 33q d 5 and [d2] + 1 q2d 33 1.     

Dynamic Changes in Peer Victimization and Adjustment Across Middle School: Does Friends’ Victimization Alleviate Distress?

Although some adolescents are chronically bullied throughout middle school, others may only experience peer victimization temporarily. This study examined the effects of time-invariant (average level) and time-varying (year-to-year) victimization experiences across middle school on adolescents’ depressive symptoms, somatic complaints, and self-blame. A key question was whether friends’ victimization buffered students from their victimization-related distress. The diverse sample (n = 5,991) was surveyed four times between sixth and eighth grade (M_age at sixth grade = 11.54 years). Three-level multilevel models revealed both time-invariant and time-varying effects of victimization on adjustment, but these maladaptive associations were attenuated when adolescents’ friends experienced more victimization across middle school. The results suggest that even temporarily victimized youth may have unmet mental health needs, and sharing social plight with friends can protect victims.     

2维液晶中向错线的拓扑结构

Using φ-mapping method and topological current theory,the topological structure of disclination lines in 2-dimensional liquid crystals is studied.By introducing the strength density and the topological current of many disclination lines,it is pointed out that the disclination lines are determined by the singulaities of the director field,and topologically quantized by the Hopf indices and Brouwer degrees,Due to the equivalence in physics of the director fields n(x) and -n(x),the Hopf indices can be integers or half-integers,representing a generalization of our previous studies of integer Hopf indices.     

Algebraic methods in plane geometry

In this article we examine the role of mappings in elementary geometry. After making some comments about the Erlangen programme initiated by Felix Klein in 1872, wherein he proposed a way of studying geometries based on the underlying transformation groups, we see how theorems like Von Aubel’s theorem and Napoleon’s theorem can be proved in an elegant manner using similarity mappings, and how some construction problems may be solved using isometries. At the end we present a recent proof by Alain Connes of “Morley’s Miracle”, based on affine transformations. Shailesh Shirali heads a Community Mathematics Center at Rishi Valley School (KFI). He has a deep interest in teaching and writing about mathematics at the high school/post school levels, with particular emphasis on problem solving and the historical aspects of the subject.     

有向线图中的(k,l)-核

1 Introduction We consider the finite digraphs that do not havemultiple arcs or loops , but pairs of opposite arcs areallowed.LetD=(V,A) be a digraph , whereVisthe set of vertices andAis the set of arcs .If there isan arc (x,y) fromxtoy,thenthe vertexxis …     

Use of context in the word recognition process by adults with a significant history of reading difficulties

We examined whether university students who report a significant history of reading difficulties (RD; n = 24) differed from university students with no history of reading difficulties (NRD; n = 31) in how sentence context affects word recognition. Experiment 1 found no differences in how congruent sentence primes or syntactic manipulations of the sentence primes affected the performance of the two groups. However, only the RD group displayed a significant inhibition effect when the target word was preceded by an incongruent sentence prime. Experiment 2 found that the groups differed in how meaning frequency of the target word and context strength of the sentence prime affected word recognition latencies. The results suggest that the RD participants’ performance is context-sensitive and better explained by interactive models of language processing than by modular models.     

The Effect of Fine Motor Skill Activities on Kindergarten Student Attention

This study explored the effect of fine motor skill activities on the development of attention in kindergarteners (n = 68) in five classes at a suburban public school in the Intermountain West through a pretest/posttest experimental group (n = 36) control group (n = 32) design. All children received the regular curriculum which included typical fine motor activities such as painting, coloring, writing, and play activities with small items. The treatment was a series of supplemental fine motor activities in which children used tongs, tweezers, and spoons to move small items. The assessment was the attention subtest of the Cognitive Assessment System (CAS) (Naglieri, J. A., & Das, J. P. (1997a). Cognitive assessment system. Itasca, IL: Riverside). A significant group × sex interaction with females positively responding to the treatment was found, suggesting that fine motor skill activities are effective in increasing female kindergartners’ attention. Further studies exploring effective materials for males and factors such as student choice and interest are needed.     

Purposely Teaching for the Promotion of Higher-order Thinking Skills: A Case of Critical Thinking

This longitudinal case-study aimed at examining whether purposely teaching for the promotion of higher order thinking skills enhances students’ critical thinking (CT), within the framework of science education. Within a pre-, post-, and post–post experimental design, high school students, were divided into three research groups. The experimental group (n = 57) consisted of science students who were exposed to teaching strategies designed for enhancing higher order thinking skills. Two other groups: science (n = 41) and non-science majors (n = 79), were taught traditionally, and acted as control. By using critical thinking assessment instruments, we have found that the experimental group showed a statistically significant improvement on critical thinking skills components and disposition towards critical thinking subscales, such as truth-seeking, open-mindedness, self-confidence, and maturity, compared with the control groups. Our findings suggest that if teachers purposely and persistently practice higher order thinking strategies for example, dealing in class with real-world problems, encouraging open-ended class discussions, and fostering inquiry-oriented experiments, there is a good chance for a consequent development of critical thinking capabilities.     

Towards global wisdom in the ERA of digitalization and communication

Conclusions Educational reform needs the synergetic efforts of UNESCO, the EU, national and local governments, policy makers, educators, business communities, public interest groups, parents, citizens, and non-governmental organizations, such as IFIP and the International Association for the Evaluation of Educational Achievements (IEA). It is a matter of crucial importance that countries in transition participate in the EU’s educational and training initiatives and programmes, such as PHARE (including TEMPUS), COPERNICUS, SOCRATES and LEONARDO, as well as in all UNESCO initiatives and projects. Original language: English Blagovest Sendov (Bulgaria) Former speaker of the Bulgarian Parliament and vice-speaker of the present Parliament. Graduate in mathematics from Sofia University, and specialist in numerical analysis from Moscow University and computer science from Imperial College, London. He has held the posts of Dean of the Mathematical Faculty and Rector of Sofia University, and president of the Bulgarian Academy of Sciences, the International Federation for Information Processing, the International Association of Universities, and vice-president of the International Council of Scientific Unions. From 1979 to 1990 leader of the Bulgarian research group on new curricula integrating school subjects with computers and information technology. This is a slightly edited version of a paper presented at UNESCO’s second International Congress on Education and Informatics: Educational Policies and New Technologies, Moscow, 1–5 July 1996.     

On the Concept of Energy: How Understanding its History can Improve Physics Teaching

Some physicists have pointed out that we do not know what energy is. Many studies have shown that the concept of energy is a problem for teaching. A study of the history of the concept shows that the discoverers of energy did not find anything which is indestructible and transformable but rather that the concept of energy underwent a change of meaning and energy was considered a substance towards the end of the nineteenth century. In distinguishing between the treatment of phenomena and the theories carried out by Mayer and Joule, it can be concluded that they established equivalences between different domains, such as motion and heat, motion and electricity or position and motion. This complies with the interpretation presented in textbooks published about a century ago and enables us to overcome some difficulties with the concept of energy.

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Literary critic versus expert commentator: The case of katharine tozer'sMumfie marches on

Nicholas Tucker taught English in comprehensive schools in London before qualifying as an educational psychologist. He is now a lecturer in developmental psychology at the University of Sussex, with a special interest both in children's reactions to literature and in the history and present-day status of childhood itself. He has written five books for children as well as books about children's literature, includingThe Child and the Book: A Literary and Psychological Exploration, reissued in 1990 by Cambridge University Press in their new Canto “Classics” series.     

How individual interest influences situational interest and how both are related to knowledge acquisition: A microanalytical investigation

The extent to which a student experiences situational interest during a learning task is dependent on at least two factors: (1) external stimuli in the learning environment that arouse interest and (2) internal dispositions, such as individual interest. The objective of the present study was to disentangle how both factors influence situational interest during task engagement. Two data sets were collected from primary school science (N = 186) and secondary school history students (N = 71). Path analysis was used to examine the influence of individual interest on seven situational interest measurements and knowledge acquisition. The results suggest that individual interest has only a significant influence on situational interest at the beginning of a task and then its influence fades. In addition, individual interest is not a significant predictor of learning. Only situational interest predicts knowledge acquisition. Implications of these findings for interest research are discussed.     

组成于平行机和批处理机的二阶段混合流水作业问题

A hybrid two-stage flowshop scheduling problem was considered which involves m identical parallel machines at Stage 1 and a burn-in processor M at Stage 2, and the makespan was taken as the minimization objective. This scheduling problem is NP-hard in general. We divide it into eight subcases. Except for the following two subcases： （1） b≥ an, max{m, B} 〈 n; （2） a1 ≤ b ≤ an, m ≤ B 〈 n, for all other subcases, their NP-hardness was proved or pointed out, corresponding approximation algorithms were conducted and their worst-case performances were estimated. In all these approximation algorithms, the Multifit and PTAS algorithms were respectively used, as the jobs were scheduled in m identical parallel machines.     

On the move: What causes animals to disperse?

The phenomenon of dispersal has been defined in many different ways, of which one of the oldest definitions was the one given by Lidicker in 1975 [1]. He defined dispersal as“…any movement of individual organisms or their propagules in which they leave their home area, sometimes establishing a new home area. This does not include short-term exploratory movements, or changes in the boundaries of a home range, such that the new range includes at least part of the former. Dispersal thus produces homeless travelers (vagrants) who are in search of a new home.” Dispersal has been at the forefront of research involving animal behaviour and ecology for a very long time.     

在n维赋范空间中单位球体积及其渐进性质

Let Bnp={x ∈ Rn‖|x ‖p≤l} be the unit ball ofp norm in the n-dimensional normed space p. The formula for the volume of Bnp was obtained and its asymptotic properties were found out as n→∞and p→∞.