Análise das técnicas mobilizadas por licenciandos em matemática ao resolverem tarefas visuaisAnalysis of techniques mobilized by mathematics degree students when solving visual tasks

ResumoFundamentado em uma pesquisa de mestrado, este trabalho tem como objetivo analisar as técnicas mobilizadas por 18 estudantes do 2º ano de Licenciatura em Matemática ao resolverem tarefas matemáticas visuais, classificando-as segundo seu nível de Visualidade Matemática e identificando, quando houver, a presença das habilidades de visualização - Interpretação da Informação Figurativa (IFI) e Processamento visual (VP), descritas por Alan J. Bishop. Os dados foram coletados por meio do registro escrito das resoluções de duas tarefas matemáticas propostas aos participantes da pesquisa. A análise dos dados revelou que das 15 resoluções apresentadas, 7 foram classificadas como técnicas parcialmente visuais, 5 como não-visuais e 3 como visuais. Todas apresentaram indícios da habilidade IFI; e 10 do VP.Palavras-chave: Visualização matemática, Tarefas matemáticas, Habilidades visuais. AbstractBased on a master's research, this work aims to analyze the techniques mobilized by 18 students of the 2nd year of a Mathematics degree course when solving visual mathematical tasks, classifying them according to their level of Mathematical Visuality and identifying, if any, the presence of visualization skills - Interpretation of Figurative Information (IFI) and Visual Processing (VP), described by Alan J. Bishop. The data were collected through the written record of the resolutions of two mathematical tasks proposed to the research participants. Data analysis revealed that of the 15 resolutions presented, 7 were classified as partially visual techniques, 5 as non-visual, and 3 as visual. All of them showed evidence of IFI skill; and 10 of the PV.Keywords: Mathematical visualization, Mathematical tasks, Visual skills.ResumenBasado en una investigación de máster, este trabajo tiene como objetivo analizar las técnicas movilizadas por 18 estudiantes de 2o curso de la carrera docente de Matemáticas a la hora de resolver tareas matemáticas visuales, clasificándolas según su nivel de Visualidad Matemática e identificando, en su caso, la presencia de Habilidades de visualización: Interpretación de Información Figurativa (IFI) y Procesamiento Visual (VP), descrito por Alan J. Bishop. Los datos fueron recolectados a través del registro escrito de las resoluciones de dos tareas matemáticas propuestas a los participantes de la investigación. El análisis de los datos reveló que, de las 15 resoluciones presentadas, 7 se clasificaron como técnicas parcialmente visuales, 5 como no visuales y 3 como visuales. Todos mostraron evidencia de habilidad de IFI; y 10 del PV.Palabras clave: Visualización matemática, Tareas matemáticas, Habilidades visuales.

Pattern Recognition and Color Modularity in Mathematics and Art

Pattern recognition is a useful tool for mathematics, mathematical visualization, and art. After a brief description of Bongard methodology in the field of pattern recognition, the author combines consequential elements of this technique and principles of color modularity to transform a five- to a ten-pointed star polygon, and in doing so attains a rich, persuasive visualization. This process that can be repeated for more complex objects brought to light additional valuable implication for mathematics visualization, visual communication, and pattern recognition practices.

Computational geometry as a tool for studying root-finding methods

We present an efficient method from Computational geometry, a branch of computer science devoted to the study of algorithms, for mathematical visualization of a third order root solver. For many decades the quality of iterative methods for solving nonlinear equations were analyzed only by using numerical experiments. The disadvantage of this approach is the inconvenient fact that convergence behavior strictly depends on the choice of initial approximations and the structure of functions whose zeros are sought, which often makes the convergence analysis very hard and incomplete. For this reason in this paper we apply dynamic study of iterative processes relied on basins of attraction, a new and powerful methodology developed at the beginning of the 21th century. This approach provides graphic visualization of the behavior of convergent sequences and, consequently, offers considerably better insight into the quality of applied root solvers, especially into the domain of convergence. For demonstration, we present dynamic study of one parameter family of Halley?s type introduced in the first part of the paper. Characteristics of this family are discussed by basins of attractions for various values of the involved parameter. Special attention is devoted to clusters of polynomial roots, one of the most difficult problems in the topic. The analysis of the methods and presentation of basins of attractions are performed by the computer algebra system Mathematica.

Assessment of student achievement and views on the impact of instruction with visualization of identities in (ax+b)n form in mathematics

The aim of the present study is to examine the impact of instruction with visualization of identities in form in mathematics on students’ achievement and opinions. The study employs an explanatory design, a mixed method, through which qualitative and quantitative data were collected and analysed. Quantitative data in the study were obtained using the Identities and Binomial Exponential Achievement Test to determine the effect of visualization-assisted instruction on students’ academic achievement. Qualitative data, regarding the student views on the subject, were collected through a structured interview form. Participants of the study, 53 students, were selected by appropriate sampling method among the first-year univeristy students. The t-test and one-way covariance analysis were used to analyse the quantitative data and the content analysis method was used for the qualitative data. The results obtained from quantitative data indicated that instruction of algebraic expressions through mathematical visualization increased student achievement in comparison to traditional instruction. On the other hand, the qualitative data indicated that visualization facilitated a positive development in instruction of algebraic expressions, rendered the teaching process to be more interesting and enjoyable and led the students to acquire conceptual learning skills instead of memorizing.