Saberes e Práticas do Professor que Ensina Matemática na Formação Continuada

ResumoEste artigo refere-se a um recorte de pesquisa de doutorado e apresenta reflexões decorrentes da formação continuada de um grupo de professores, com diferentes titulações em Pedagogia e Matemática, sobre como pensam os problemas matemáticos e os veiculam em sala de aula. A trajetória investigativa contou com uma metodologia qualitativa e um papel de intervenção. Os dados coletados foram utilizados para compreender as reflexões dialógicas que dimensionaram o processo de treinamento e para analisar os registros obtidos pelos protocolos de atividades. Os estudos teóricos de Llinares, Shulman, Zeichner, Vergnaud, Ma e Ball Thames e Phelps orientaram a ação formativa e favoreceram a análise reflexiva. As análises demonstraram a importância de reconhecer a competência profissional de cada um e entender como suas atividades formativas se inter-relacionam e se complementam. Conclui-se que a ação formadora explora a diversidade de momentos e promove a constituição de grupos distintos, propicia uma aprendizagem diferenciada e a redefinição das competências profissionais dos professores que ensinam Matemática. Palavras chave: Problemas Aditivos. Projeto Observatório. Estrutura Multiplicativa. Aprendizagem Matemática. AbstractThis article refers to an excerpt from doctoral research and presents reflections resulting from the continued formation of a group of teachers, with different degrees in Pedagogy and Mathematics, about how they think about mathematical problems and convey them in the classroom. The investigative trajectory counted on a qualitative methodology and an intervention role. The data collected was used to understand the dialogical reflections that dimensioned the training process and to analyze the records obtained by the activity protocols. The theoretical studies of Llinares, Shulman, Zeichner, Vergnaud, Ma, Ball, Thames and Phelps guided the formative action and favored reflective analysis. The analyzes demonstrated the importance of recognizing each other's professional competence and understanding how their educational activities interrelate and complement each other. It is concluded that the formation-action explores the diversity of moments and promotes the constitution of distinct groups, provides a differentiated learning and the redefinition of the professional competences of teachers who teach Mathematics. Keywords: Additive Problems. Observatory Project. Multiplicative Structure. Mathematical Learning.

SUM-PRODUCT ESTIMATES FOR DIAGONAL MATRICES

Given $d\in \mathbb{N}$ , we establish sum-product estimates for finite, nonempty subsets of $\mathbb{R}^{d}$ . This is equivalent to a sum-product result for sets of diagonal matrices. In particular, let $A$ be a finite, nonempty set of $d\times d$ diagonal matrices with real entries. Then, for all $\unicode[STIX]{x1D6FF}_{1}<1/3+5/5277$ , $$\begin{eqnarray}|A+A|+|A\cdot A|\gg _{d}|A|^{1+\unicode[STIX]{x1D6FF}_{1}/d},\end{eqnarray}$$ which strengthens a result of Chang [‘Additive and multiplicative structure in matrix spaces’, Combin. Probab. Comput.16(2) (2007), 219–238] in this setting.

Dynamical Semigroups in the Birkhoff Polytope of Order 3 as a Tool for Analysis of Quantum Channels

In the present paper we show a link between bistochastic quantum channels and classical maps. The primary goal of this work is to analyse the multiplicative structure of the Birkhoff polytope of order 3 (the simplest nontrivial case). A suitable complex parametrization of the Birkhoff polytope is proposed, which reveals several its symmetries and characteristics, in particular: (i) the structure of Markov semigroups inside the Birkhoff polytope, (ii) the relation between the set of Markov time evolutions, the set of positive definite matrices and the set of divisible matrices. A condition for Markov time evolution of semigroups in the set of symmetric bistochastic matrices is derived, which leads to an universal conserved quantity for all Markov evolutions. Finally, the complex parametrization is extended to the Birkhoff polytope of order 4.