On pythagorean triples of the form (i, i + l, k)

A Pythagorean triple is a triad of positive integers (x, y, z) which satisfy the Pythagoras’ equation x ² + y ² = z ². In this article, we shall consider triples of the form (i, i + 1, k), and the recurrence relations governing them. In the process, we also solve completely the equation i ²+ (i + 1)² = k ².     

Applied mathematical problem solving,modelling, applications,and links to other subjects — State,trends and issues in mathematics instruction

The paper will consist of three parts. In part I we shall present some background considerations which are necessary as a basis for what follows. We shall try to clarify some basic concepts and notions, and we shall collect the most important arguments (and related goals) in favour of problem solving, modelling and applications to other subjects in mathematics instruction. In the main part II we shall review the present state, recent trends, and prospective lines of development, both in empirical or theoretical research and in the practice of mathematics instruction and mathematics education, concerning (applied) problem solving, modelling, applications and relations to other subjects. In particular, we shall identify and discuss four major trends: a widened spectrum of arguments, an increased globality, an increased unification, and an extended use of computers. In the final part III we shall comment upon some important issues and problems related to our topic.Written version of a Survey Lecture given jointly at the Sixth International Congress on Mathematical Education (ICME-6, Budapest 1988). A condensation was published in a volume with contributions from the ICME-6 Theme Groups on Problem Solving, Modelling and Applications and on Mathematics and Other Subjects (Blum/Niss/Huntley 1989, pp. 1–21).     

Quelques Erreurs Pouvant être Liées à Une Difficulté à Concevoir un Ensemble Comme un Objet Distinct De Ses éléments Chez Des étudiants Et Des étudiantes Universitaires

As a first step of a project on learning difficulties in elementary set theory, we carried out a study aimed at identifying some of the difficulties encountered by a group of 21 mathematics majors taking a course on logic and set theory. In this paper, we shall discuss one of the difficulties that we observed, namely a difficulty in conceiving a set as an object distinct from its elements, i.e. failing to fully grant sets the status of objects. This difficulty may be related to the following three types of errors that we observed in the students' first assignment: (1) confusing belonging and inclusion, (2) confusing the union of sets A, B, C, ... with the set whose elements are A, B, C, … and (3) adding or deleting curly brackets. We shall present excerpts from the students' work illustrating these three types of errors and explain how they may be related to a difficulty in conceiving a set as an object distinct from its elements.     

A STRUCTURAL MODEL OF PARENT INVOLVEMENT WITH DEMOGRAPHIC AND ACADEMIC VARIABLES

Parental involvement is well documented as a significant contributor to the self‐efficacy and academic achievement of students. A structural equation model of parent involvement with family socioeconomic status, student gender, parents’ aspirations for their children, mathematics efficacy, and mathematics achievement was tested to examine whether parent involvement in the 10th grade remains relevant to achievement. A sample of data pertaining to 8,673 10th graders from the Educational Longitudinal Study was analyzed. The results indicated that the fit of the measurement model to the data was good (χ² = 3081.62, df = 87, p = .0, normed fit index [NFI] = .96, comparative fit index [CFI] = .96, root mean square error of approximation [RMSEA] = .064), as was the structural model (χ² = 3470.69, df = 94, p = .00, NFI = .96, CFI = .96, RMSEA = .065). Although the effect was small in magnitude, parent involvement in advising had a significant indirect relationship with mathematics achievement via mathematics efficacy of 10th graders.     

On a Problem of Alaoglu and Erdős

Starting with an elementary problem that appeared in the Putnam mathematics competition, we proceed to discuss some techniques of transcendental number theory and prove the following result. If p, q, r are distinct primes and if c is a real number with the property that p^c, q^c, r^c are integers, then c must be a non-negative integer. The tools used are some linear algebra and complex analysis. The zero-density estimate method discussed here was used by Alan Baker to prove his celebrated theorem on linear forms in logarithms. The question as to whether we can replace three primes by two primes is an open question.     

Hands-on mathematics: two cases from ancient Chinese mathematics

In modern mathematical teaching, it has become increasingly emphasized that mathematical knowledge should be taught by problem-solving, hands-on activities, and interactive learning experiences. Comparing the ideas of modern mathematical education with the development of ancient Chinese mathematics, we find that the history of mathematics in ancient China is an abundant resource for materials to demonstrate mathematics by hands-on manipulation. In this article I shall present two cases that embody this idea of a hands-on approach in ancient Chinese mathematics, at the same time offering an opportunity to show how to utilize materials from the history of Chinese math in modern mathematical education.

Linking mathematics with engineering applications at an early stage – implementation,experimental set-up and evaluation of a pilot project

Too difficult, too abstract, too theoretical – many first-year engineering students complain about their mathematics courses. The project MathePraxis aims to resolve this disaffection. It links mathematical methods as they are taught in the first semesters with practical problems from engineering applications – and thereby shall give first-year engineering students a vivid and convincing impression of where they will need mathematics in their later working life. But since real applications usually require more than basic mathematics and first-year engineering students typically are not experienced with construction, mensuration and the use of engineering software, such an approach is hard to realise. In this article, we show that it is possible. We report on the implementation of MathePraxis at Ruhr-Universität Bochum. We describe the set-up and the implementation of a course on designing a mass damper which combines basic mathematical techniques with an impressive experiment. In an accompanying evaluation, we have examined the students' motivation relating to mathematics. This opens up new perspectives how to address the need for a more practically oriented mathematical education in engineering sciences.     

Relationship of creativity and critical thinking to pattern recognition among Singapore private school students

Abstract

Cognitive pattern recognition is known to be an important skill for academic subjects such as mathematics, science, languages, or even humanities. In this study, we investigate the relationships between creativity, critical thinking, and pattern recognition among 203 private school students in Singapore. The instruments used include a creativity test (modified Creativity Selected Elements Questionnaire), a Critical Thinking Test (modified Cornell Critical Thinking), and a pattern recognition test. The main data analysis is done using the SMART-PLS structural equation modeling software. The results of the study reveal that creativity is a weak predictor of pattern recognition (β?=?0.131, p?>?0.05, f² = 0.024) but critical thinking is a good predictor (β?=?0.517, p?<?0.05, f² = 0.374). An implication of the research outcome is that more training on critical thinking should be given to the students to improve their pattern recognition ability.     

Mathematical contributions of Archimedes: Some nuggets

Archimedes is generally regarded as the greatest mathematician of antiquity and alongside Isaac Newton and C F Gauss as the top three of all times. He was also an excellent theoreticiancum-engineer who identified mathematical prob lems in his work on mechanics, got hints on their solution through engineering techniques and then solved those mathematical problems, many a time discovering fundamental results in mathematics, for instance, the concepts oflimits andintegration. In his own words,“… which I first dis covered by means of mechanics and then exhibited by means of geometry”. In this article we briefly describe some of his main contributions to mathematics.     

India’s arrival on the modern mathematical scene

Modern mathematics, and modern science in general, was embraced enthusiastically in the Indian subcontinent quite early, thanks to a large extent to the tradition of learning going back to the ancient times. By the early decades of the 20th century the Indian mathematical community had made important research contributions in diverse areas of mathematics, including number theory, real and complex analysis, diffrential geometry, diffrential equations, algebra, combinatorial mathematics and applied mathematics. Interaction with world leaders in the field, reforms in the educational system, establishment of societies for actively pursuing study and discussion of mathematics, publication of mathematical journals, were some of the major progressive steps taken. Apart from the indigenous leadership, participation of a few enlightened foreigners in the process also paved the way for the advancement of the subject in the country. We recount here the story of the early developments of the mathematical scene in India, and of the various players involved.     

Using History to Teach Mathematics: The Case of Logarithms

Many authors have discussed the question why we should use the history of mathematics to mathematics education. For example, Fauvel (For Learn Math, 11(2): 3–6, 1991) mentions at least fifteen arguments for applying the history of mathematics in teaching and learning mathematics. Knowing how to introduce history into mathematics lessons is a more difficult step. We found, however, that only a limited number of articles contain instructions on how to use the material, as opposed to numerous general articles suggesting the use of the history of mathematics as a didactical tool. The present article focuses on converting the history of logarithms into material appropriate for teaching students of 11th grade, without any knowledge of calculus. History uncovers that logarithms were invented prior of the exponential function and shows that the logarithms are not an arbitrary product, as is the case when we leap straight in the definition given in all modern textbooks, but they are a response to a problem. We describe step by step the historical evolution of the concept, in a way appropriate for use in class, until the definition of the logarithm as area under the hyperbola. Next, we present the formal development of the theory and define the exponential function. The teaching sequence has been successfully undertaken in two high school classrooms.     

关于Pell方程x^2-5py^2=-1

运用初等数论的方法证明了Pell方程x^2-5py^2=-1有正整数解.这里p〉3且p是fermat素数.     

Let Lakatos Be! A Commentary on “Would the Real Lakatos Please Stand Up”

In this commentary, some remarks are offered on David Pimm, Mary Beisiegel, and Irene Meglis’ article “Would the Real Lakatos Please Stand up.” The commentary focuses on relatively recent developments in the philosophy of mathematics based on the work of Lakatos; on theory development in mathematics education; and offers critique on whether Lakatos’ Proofs and Refutations (1976) can be directly implicated in mathematics education.

“Nature and nature’s laws lay hid in night; God said, Let Newton be! and all was light.” (Pope, 1688-1744¹)

     

拟线性中立型差分方程 k-单调正解存在显著的充分条件

本文研究了拟线性中立型差分方程 △(x_n-x_(n-k)~α+q_nX_(n-1)~α=0 n=0,1,2,….正解存在性,其中q_n≥0,n=0,l,2…,k>0和1都是非负整数,α为两正奇数之商。获得了该方程存在k-单调正解一些显著的充分条件。     

线性方程组理论在初等数学教学中的应用

在平面与空间解析几何中,利用线性方程组理论确定两条直线之间的位置关系,并且利用齐次线性方程组的一个结论求出二元方程组的解,此外它还可以判别一元多次方程的重根,从而将大学所学的高等数学知识应用到初等数学中,为在初等数学中应用高等数学知识打下一个铺垫.     

几个不定方程在Q（√-11）和Q（√-43）中的解

根据代数数论的理论,将初等数论中的一些结论推广到更大的代数整数环中,应用这些结论确定了几个著名的不定方程在虚二次域的整数环中的解,指出了费尔玛方程在比整数环更大的环中也没有非平凡解。     

PREDICTING MATHEMATICS ACHIEVEMENT: THE INFLUENCE OF PRIOR ACHIEVEMENT AND ATTITUDES

Achievement in mathematics is inextricably linked to future career opportunities, and therefore, understanding those factors that influence achievement is important. This study sought to examine the relationships among attitude towards mathematics, ability and mathematical achievement. This examination was also supported by a focus on gender effects. By drawing on a sample of Australian secondary school students, it was demonstrated through the results of a multivariate analysis of variance that females were more likely to hold positive attitudes towards mathematics. In addition, the predictive capacity of prior achievement and attitudes towards mathematics on a nationally recognised secondary school mathematics examination was shown to be large (R ²  =  0.692). However, when these predictors were controlled, the influence of gender was non-significant. Moreover, a structural equation model was developed from the same measures and subsequent testing indicated that the model offered a reasonable fit of the data. The positing and testing of this model signifies growth in the Australian research literature by showing the contribution that ability (as measured by standardised test results in numeracy and literacy) and attitude towards mathematics play in explaining mathematical achievement in secondary school. The implications of these results for teachers, parents and other researchers are then considered.     

Theorema aureum — 2

In the first part of this article, we had introduced the notion of quadratic reciprocity and dwelt briefly on its history, which goes back all the way to the work of Fermat. Then we discussed the Law of Quadratic Reciprocity (‘QRL’), which Gauss named Theorema Aureum. Following this, we gave a not too well known proof of the QRL, due to G Rousseau. Now we give two more proofs of the QRL, drawing respectively from ideas in linear algebra and field extensions; they too are not very well known. Shivam Kumar graduated from Indian Statistical Institute, Bangalore and is joining the London School of Economics for MSc in applicable mathematics. His interest lies in expanding the existing applied paradigm of mathematics from stock market to unchartered subjects such as sociometrics.     

Expectancy-value and cognitive process outcomes in mathematics learning: a structural equation analysis

Existing research has yielded evidence to indicate that the expectancy-value theoretical model predicts students' learning in various achievement contexts. Achievement values and self-efficacy expectations, for example, have been found to exert positive effects on cognitive process and academic achievement outcomes. We tested a conceptual model that depicted the interrelations between the non-cognitive (task value, self-efficacy) and cognitive (deep-learning approach, reflective-thinking) processes of learning, and academic achievement outcomes in mathematics. University students (n = 289) were administered a number of Likert-scale inventories and LISREL 8.80 was used to test various a priori and a posteriori models. Structural equation modeling yielded some important findings: (1) the positive temporally displaced effects of prior academic achievement, self-efficacy expectations and task value on achievement in mathematics, (2) the positive relations between self-efficacy expectations and task values and cognitive process outcomes and (3) the possible mediating role of self-efficacy expectations and task value between prior academic achievement and deep learning, reflective-thinking practice and academic achievement. Overall, our research investigation has provided empirical groundings for further advancement into this area of students' learning.     

Intuitive rules in science and mathematics: the case of ‘more of A ‐‐ more of B’

In the last twenty years researchers have studied students’ mathematical and scientific conceptions and reasoning. Most of this research is content‐specific. It has been found that students often hold ideas that are not in line with accepted scientific notions. In our joint work in mathematics and science education it became apparent that many of these alternative conceptions hail from the same intuitive rules. We have so far identified two such rules: ‘The more of A, the more of B’ and, ‘Everything can be divided by two’. The first rule is reflected in students’ responses to many tasks, including all classical Piagetian conservation tasks (conservation of number, area, weight, volume, matter, etc.), in all tasks related to intensive quantities (density, temperature, concentration, etc.), and in tasks related to infinite quantities. The second rule is observed in responses related to successive division of material and geometrical objects, and in successive dilution tasks. In this paper we describe and discuss the first rule and its relevance to science and mathematics education. In a second paper (Tirosh and Stavy, in press) we shall describe and discuss the second rule.