Jaime Carvalho e Silva

The political context

People have studied, learned and used mathematics for over four thousand years. Decisions on what is to be taught in schools, and how, are ultimately political, influenced by a number of factors including the experience of teachers, expectations of parents and employers, and the social context of debates about the curriculum. The ICMI study is posited on the experience of many mathematics teachers across the world that its history makes a difference: that having history of mathematics as a resource for the teacher is beneficial.

Philosophical, multicultural and interdisciplinary issues

School mathematics reflects the wider aspect of mathematics as a cultural activity. From the philosophical point of view, mathematics must be seen as a human activity both done within individual cultures and also standing outside any particular one. From the interdisciplinary point of view, students find their understanding both of mathematics and their other subjects enriched through the history of mathematics. From the cultural point of view, mathematical evolution comes from a sum of many contributions growing from different cultures.

A complete comparison of 25 contraction conditions

Abstract In this paper we establish a comparison of the 25 contraction conditions considered in a previous paper of Rhoades [16], thus completing the comparison first made in a systematic way in that paper.

An effective algorithm for computing all-terminal reliability bounds

The exact calculation of all-terminal reliability is not feasible in large networks. Hence estimation techniques and lower and upper bounds for all-terminal reliability have been utilized. Here, we propose using an ordered subset of the mincuts and an ordered subset of the minpaths to calculate an all-terminal reliability upper and lower bound, respectively. The advantage of the proposed new approach results from the fact that it does not require the enumeration of all mincuts or all minpaths as required by other bounds. The proposed algorithm uses maximally disjoint minpaths, prior to their sequential generation, and also uses a binary decision diagram for the calculation of their union probability. The numerical results show that the proposed approach is computationally feasible, reasonably accurate and much faster than the previous version of the algorithm. This allows one to obtain tight bounds when it not possible to enumerate all mincuts or all minpaths as revealed by extensive tests on real-world networks.

What International Studies Say about the Importance and Limitations of Using Computers to Teach Mathematics in Secondary Schools

The use of technology in schools has been one of the most debated topics around mathematics education. In some countries there is a huge investment, in others there is a downscaling. Malaysia decided in 2013 to put its 10 million students to use Google laptops and Google apps, while Australia in the same year decided it would not continue funding their own high school laptop program. Who is right from the educational point of view? The last major curriculum document written in the world to date, the Common Core State Standards-CCSS in the United States, whose mathematics part is coordinated by the well known mathematician William McCallum, sets as one of its standards for mathematical practice: “Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software.” Strong moves need substantiation from research, including the analysis of the existing situation in different countries. What does research say about the use of computers in schools in present time and the use of different pieces of software from spreadsheets to computer algebra systems?

An algorithm for computing all-terminal reliability bounds

The exact calculation of all-terminal reliability is not feasible in large networks. Hence estimation techniques and lower and upper bounds for all-terminal reliability have been utilized. We propose using an ordered subset of the mincuts and an ordered subset of minpaths to calculate an all-terminal reliability upper and lower bound, respectively. The advantage of the proposed approach results from the fact that it does not require the enumeration of all mincuts or all minpaths as required by other bounds. The performance of the algorithm is compared with the first two Bonferroni bounds, for networks where all mincuts could be calculated. The results show that the proposed approach is computationally feasible and reasonably accurate. Thus allowing one to obtain bounds when it not possible to enumerate all mincuts or all minpaths.

The generalized Goursat-Darboux problem for a third order operator

It is proved that if a generalized Goursat-Darboux problem is C∞well posed then the operator cannot contain derivatives with respect to one of the variables.

Gevrey Well Posedness of Goursat-Darboux Problems and Asymptotic Solutions

We consider the generalized Goursat-Darboux problem for a third order linear PDE with real constant coefficients. Our purpose is to find necessary conditions for the problem to be well-posed in the Gevrey classes. Since this problem can be reduced to the Cauchy problem using permutations of independent variables, we solve it for a ODE with complex coefficients and two unknown initial data. In order to prove our results, we first construct an explicit solution of a family of problems with initial data depending on a parameter \(\eta > 0\) and then we obtain an asymptotic representation of a solution as \(\eta \) tends to infinity.

Fatores influenciadores dos resultados de matemática de estudantes portugueses e brasileiros no PISA: revisão integrativa

The present integrative review has the goal of knowing the main factors influencing the results of Portuguese and Brazilian students in the editions of PISA, from 2000 to 2012, according to the published literature. Five factors and ten sub-factors have been identified. The more determinant factors in the considered studies were the educational system, the socioeconomic context of the students and the characteristics of schools. This integrative review gives evidence of the multiplicity of factors that affect the performance of students in PISA, in Portugal as in Brazil. There also exists a significant gap in studies in the literature, as there are factors that are scarcely or not at all analysed in both countries. There is in consequence a need of a bigger investment in research in this area, taking advantage of the huge set of data generated by PISA.

On the product of the slopes of two perpendicular lines

We show that when one line approaches a vertical line and another line approaches a horizontal line, the product of the slopes can approach any real number or +oo or — oo or nothing, thus clarifying preceding statements of Subramaniam and Das. We stress that the slope of a vertical line does not exist and explain why all these discussions are very useful in the classroom.